MATH13 Coursewares

163
Lesson 1.1: POLYGON Lesson 1.2 Triangles Lesson 1.3 Quadrilaterals Week 1 and Week 2 Math 13 Solid Mensuration

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Lesson 1.1: POLYGON Lesson 1.2 Triangles Lesson 1.3 QuadrilateralsWeek 1 and Week 2 Math 13 Solid Mensuration‡ A polygon is a closed plane figure that is joined by line segments. ‡ A polygon may also be defined as a union of line segments such that: i) each endpoint is the endpoint of only two segments; ii) no two segments intersect except at an endpoint; and iii) no two segments with the same endpoint are collinear.Reference: Solid Mensuration by Richard EarnhartParts of a PolygonSide

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Lesson 1.1: POLYGONLesson 1.2 Triangles

Lesson 1.3 Quadrilaterals

Week 1 and Week 2Math 13

Solid Mensuration

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• A polygon is a closed plane figure that is joined by line segments.

• A polygon may also be defined as a union of line segments such that: i) each endpoint is the endpoint of only two segments; ii) no two segments intersect except at an endpoint; and iii) no two segments with the same endpoint are collinear.

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Parts of a Polygon

Side or Edge

Exterior Angle

Vertex

Diagonal

Interior Angle

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Types of Polygon•Regular Polygon.In a regular polygon, all angles are equal and all sides are of the same length. Regular polygons are both equiangular and equilateral.•Equiangular Polygon.A polygon is equiangular if all of its angles are congruent.•Equilateral Polygon.A polygon is equilateral if all of its sides are equal.•Irregular Polygon.A polygon that is neither equiangular nor equilateral is said to be an irregular polygon.

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NAMING OF POLYGON

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Pythagorean Identities

Negative Arguments Identities

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• For numbers from 100 to 999, we construct the name of the polygon by starting with the prefix for the hundreds digit taken from the ones digit minus the “gon” followed by "hecta," then proceed as before.

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By ratio and proportion,

Similar Polygons We say that two polygons are similar if their corresponding interior angles are congruent and their corresponding sides are proportional.

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The altitude a of the triangle is called the apothem The angle that is opposite the base of this triangle is called the central angle .

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Examples

Perimeter:

Central Angle:

Apothem:

n = no. of sides

s/2

θ/2a

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No. of Diagonals :

Interior Angle:

Sum of Interior Angle:

AREA

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Example 1, page 8Find the area of a regular nonagon with a side that measures 3 units. Also find the number of diagonals and the sum of its interior angles.ANS: A= 55.64 s.u., D=27, SIA=1260°

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5. Find the sum of the interior angle of a regular triacontakaitetragon.7. Name each polygon with the given number of sides. Also find the

corresponding number of diagonals.a) 24b) 181c) 47d) 653

11. The number of diagonals of a regular polygon is 35. Find the area of the polygon if its apothem measures 10 cm.

12. The number of diagonals of a regular polygon is 65. Find perimeter of the

polygon if its apothem measures 8 in. 13. The sum of interior angles of a regular polygon is 1260° . Find the area of

the polygon if the perimeter is 45 cm.

EXERCISES 1.1 pp9-11

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Homework 1.1

• Nos. 15, 17, 19, 21, 23 & 25 pp 11-12Solid Mensuration by Earnhart

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Similar Triangles: • Corresponding angles are congruent and the corresponding

sides are proportional.• same shape, different size, different measurement but in

proportion.

1.2 TRIANGLES

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Lines Connected with Triangles

An altitude of a triangle is the line segment drawn from a vertex of the triangle perpendicular to the opposite side.

A median of a triangle is the line segment connecting the midpoint of a side and

the opposite vertex. An angle bisector of a triangle is the line segment

which divides an angle of the triangle into two congruent angles and has endpoints on a vertex and the opposite side. Reference: Solid Mensuration by Richard Earnhart

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• A perpendicular bisector of a side of a triangle is the line segment which meets the side at right angle and divides the side into two congruent segments.

Types of Triangle Centers• Orthocenter is the point of intersection of the

triangle’s altitudes. • The centroid is the point of intersection of the

three medians of the triangle.Reference: Solid Mensuration by Richard Earnhart

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• The incenter is the point of intersection of the three angle bisectors of the triangle.

• The circumcenter is the point of intersection of the perpendicular bisectors of the three sides of the triangle.

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Formulas for the Lengths of Altitude, Median and Angle Bisector of a Triangle

• Consider an arbitrary triangle with sides and , and angles and . Let and be the lengths of the altitude, median and bisector originating from vertex .

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• General Formula:

• SAS (Side-Angle-Side) Formula:• Heron’s Formula for SSS (Three Sides) Case:

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EXAMPLE 4: Page 17Given a triangle ABC in which the sides are AB = 30 in., AC = 50 in, and BC = 60 in. On the side AB is a point D through which a line DE is drawn and connected through a point E on side AC so that the angle AED is equal to angle ABC . If the perimeter of the triangle ADE is equal to 56 in , find the sum of the lengths of the line segments BD and CE .

ANS: 48 in Reference: Solid Mensuration by Richard Earnhart

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EXERCISES 1.2#3, p20: Find the altitude and the area of an equilateral

triangle the side of which is 8 cm.#4, p20: One side of an isosceles triangle is 10 units and

the perimeter is 42 units. Find the area of the triangle.#5, p20: Find the area of an equilateral triangle the

altitude of which is 5 cm.#7,p21: The base of an isosceles triangle and the altitude

dropped on one of the congruent sides are equal to 18 cm and 15 cm respectively. Find the sides of the triangle.

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#8,p21: Two altitudes of an isosceles triangle are equal to 20 cm and 30 cm . Determine the base angles of the triangle.

#12,p21: Find the area of a triangle with two sides that measure 6in and 9in , and the bisector of the angle between them is .

#13,p21: In an acute triangle ABC , the altitude AD is drawn. Find the area of triangle ABC if AB=15 in, AC = 18in, and BD=10in .

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Homework 1.2

• Nos. 9, 11, 15, 17, 21 pp.21-22

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1.3 Quadrilaterals

• A quadrilateral, also known as tetragon or quadrangle, is a general term for a four-sided polygon.

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• A parallelogram is a quadrilateral in which the opposite sides are parallel.

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• Parallelograms have the following important properties:

• Opposite sides are equal.• Opposite interior angles are congruent • Adjacent angles are supplementary. • A diagonal divides the parallelogram into two

congruent triangles • The two diagonals bisect each other.Reference: Solid Mensuration by Richard Earnhart

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FORMULAS

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AREA OF PARALLELOGRAM

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• A rectangle is essentially a parallelogram in which the interior angles are all right angles.

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FORMULAS

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• A square is a special type of a rectangle in which all the sides are equal.

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Formulas

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• A rhombus is a parallelogram in which all sides are equal.

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Formulas

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A trapezoid is a quadrilateral with one pair of parallel sides.

• If the non-parallel sides are congruent, the trapezoid is called an isosceles trapezoid.

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• A trapezoid which contains two right angles is called a right trapezoid.

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Area of Trapezoid

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• A trapezium is a quadrilateral with no two sides that are parallel.

• A and C are any two opposite interior angles• s is the semi-perimeter.

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• Example 8, p31• The diagonal of a square is 12 units. What is the

measure of one side of the square? Find its area and perimeter. ANS. A=72, P=

• Example 10, p32• If ABCD is a rhombus, AC=4 , and ADC is an

equilateral triangle, what is the area of the rhombus?

• ANS: A=13.86

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Example 12, p33: Find the area and the perimeter of the right trapezoid shown in the figure. ANS A=49.4, P=30.2

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EXERCISES 1.3#1, p38: The diagonal of a rectangle is 25 meters long

and makes an angle of 360 with one side of the rectangle. Find the area and the perimeter of the rectangle.

#4, p38: A rectangle and a square have the same area. If the length of the side of the square is 6 units and the longest side of the rectangle is 5 more than the measure of the shorter side. Find the dimensions of the rectangle.

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#8, p38: The area of an isosceles trapezoid is 246 m2. If the height and the length of one of its congruent sides measure 6m and 10m respectively, find the two bases.

#10, p39: A piece of wire of length 52 m is cut into two parts. Each part is then bent to form a square. It is found that the combined area of the two squares is 109 m2. Find the sides of the two squares.

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#11, p39: A rhombus has diagonals of 32 and 20 inches. Find the area and the angle opposite the longer diagonal.

# 26, p40: Find the area of a rhombus in which one side measures 10cm and one of the diagonals measures 12cm .

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Homework 1.3

• Nos. 7, 9, 15, 20, 23, 25, 28 & 29 pp.38-40.

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Lesson 2.1: CIRCLESLesson 2.2 MISCELLANEOUS

PLANES

Week 3 and Week 4Math 13

Solid Mensuration

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2.1 CIRCLES

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• A circle is a set of points, each of which is equidistant from a fixed point called the center.

• The line joining the center of a circle to any points on the circle is known as the radius.

• An arc is a portion of a circle that contains two endpoints and all the points on the circle between the endpoints.

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• By choosing any two points on the circle, two arcs will be formed; a major arc (the longer arc), and a minor arc (the shorter one).

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• A chord is a line segment joining any two points on the circle. The chord that passes through the center of the circle is called the diameter of a circle.

• A chord divides the circle into two regions, the major segment and the minor segment.

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• A sector is the figure formed by two radii and an included arc. The central angle is the angle in which the vertex lies at the center of the circle and which sides are the two radii.

• The inscribed angle is the angle in which the vertex lies on the circle and which two sides are chords of the circle.

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If circles of different radii have common center then they are referred to as concentric circles.

The region bounded by any two concentric circles is known as the annulus.

A line in the same plane as the circle is a tangent line of the circle if it intersects the circle at exactly one point on the circle.

A line is called a secant line if it intersects the circle at two points on the circle.

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• A cyclic quadrilateral is a four-sided figure inscribed in a circle, with each vertex (corner) of the quadrilateral touching the circumference of the circle.

• The sum of the opposite angles of such a quadrilateral is .

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Theorems on Circles

• If two chords intersect at a point inside the circle, then the product of the segments of one chord is equal to the product of the segments of the other chord.

• If two secant lines of a circle intersect at an exterior point, then the product of lengths of the entire secant line and its external segment is equal to the product of the lengths of the other secant line and its external segment.

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• If a tangent line and a secant line of a circle intersect at a point exterior to the circle, then the product of the lengths of the secant line and its external segment is equal to the square of the length of the tangent line.

• Every tangent line of a circle is perpendicular to the radius of the circle drawn through the point of tangency.

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• The diameter that is perpendicular to a chord bisects the chord and its two arcs.

• Conversely, the diameter that bisects a chord is perpendicular to the chord.

• Consequently, the perpendicular bisector of a chord is the diameter of the circle which must pass through the center of the circle.

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• The line of centers of two tangent circles passes through the point of tangency.

• An inscribed angle is measured by one-half of its intercepted arc.

• The angle formed by constructing lines from the ends of the diameter of a circle to a point on the circle is a right angle.

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Formulas on Circles

• Circu• Arc length: • Area of Circle:

• Area of Segment:

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• Area of Annulus region:

A polygon is inscribed in a circle if the vertices of the polygon lie on the circle. If the polygon is regular, then the measure of its side can be determined by the cosine law.

r=radius, s=side of polygon, ϴ= central angle

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• Thepolygon is circumscribed about the circleif each side of the polygon is tangent to the circle.

• if the polygon is regular, then the radius of the circle is equal to the apothem of the polygon.

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Radius of the Circle Circumscribing a Triangle

where A is the area of the triangle and .Radius of the Circle Inscribed in a Trinagle

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Area of a Cyclic Quadrilateral

Where , semi-perimeter of the quadrilateral.

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Examples

1. p54: What is the area of a circle with a circumference of cm? ANS. 254.47 sq.cm.

3. p55. A circle which has an area of 144π cm2 is cut into two segments by a chord that is 6 cm from the center of the circle. What is the area of the smaller segment? ANS. 88.45 sq.cm.

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• 4.p55 : The tangent AB and the secant BCD are drawn to a circle from the same exterior point B. If the length of the tangent is 8 inches and the external segment of the secant is 4 inches, then what is the length of the secant?

• ANS. 16 in.

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• 6, p56: An equilateral triangle is inscribed in a circle with an area equal to square units. Find the area of the triangle. ANS. 105 sq.cm

Exercise 2.1, #27, p62: A circle has an area equal to cm2. Its diameter AB coincides with one of the sides of triangle ACB in which the vertex C lies on the circle. If the triangle has an area equal to 11 cm2, find its perimeter.ANS: 22 cm

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Exercises 2.1

#9, p60: An equilateral triangle is circumscribed about a circle of radius 10 cm. What is the perimeter of the triangle? ANS 180 cm

#13: Determine the area of the segment of a circle if the length of the chord is 15 inches and located 5 inches from the center of the circle.ANS. 42.2 sq.in.

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#16: Given a circle in which the diameter AB equals 4cm. If two points C and D lie on the circle and and , find the length of the major arc CD.•

• NS

ANS. 5.53 sq.cm.

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27. p62: A circle has an area equal to cm2. Its diameter AB coincides with one of the sides of triangle ACB in which the vertex C lies on the circle. If the triangle has an area equal to 11 cm2, find its perimeter.32., p63 : Find the area of a cyclic quadrilateral with two sides that measure and units and one diagonal coincides with a diameter of the circle, the radius of which is units.•

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• A star is a star-like figure which generally consists of a polygon with triangles on its sides.

• It is a regular star if the polygon involved is a regular polygon

• The pentagram, also known as German or witch star, is a five-pointed regular star. The hexagram which is also known as David’s star or Solomon’s seal is a six-pointed regular star.

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Area of Ellipse: a=major segment, b=minor segment

Area of Parabola: b=base, a=altitude

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• Simpson’s Rule For Irregular shaped figure. If ( is even) are the lengths of a series of parallel chords of uniform interval d then the area of the figure enclosed above is given approximately by the following formula..

I

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EXERCISES• In a circle with diameter of 20 cm, a regular

five-pointed star touching its circumference is inscribed. Find the area of the star.

• What is the area of a section bounded by a closed elliptical figure in which the major and minor segments measure 60 cm and 45 cm respectively?

• What is the area of a parabola inscribed in a rectangle 30 cm long and 22 cm wide?

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HOMEWORK No.2

Exercise 2.1: #5, 7, 15, 23, 29, 33 pp. 59-63

Exercise 2.2: # 8, 10 & 12 pp. 71-72

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Lesson 3 POLYHEDRONS

Week 5Math 13

Solid Mensuration

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Solid Mensuration by EarnhartReference: Solid Mensuration by Richard Earnhart

• Dihedral Angles• The dihedral angle is the angle formed

between two intersecting planes. In the figure shown, the two planes are called faces of the dihedral angle, and the line of intersection between the planes is called the edge of the angle.

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Polyhedral Angles• A polyhedral angle is the angle formed by three or

more planes which meet at a common point. • The common point is called the vertex of the angle.

The intersecting planes are the faces of the polyhedral angle. The lines of intersection of these faces are called the edges. A plane which cuts all the faces of a polyhedral angle (except at the vertex) is called a section.

• A face angle is the angle at the vertex and formed by any two adjacent edges. A dihedral angle of the polyhedral angle is the dihedral angle formed by any two intersecting faces.

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• A convex polyhedral angle is a polyhedral angle in which any section is a convex polygon.

Important Facts:• The sum of any two face angles of a trihedral

angle is greater than the third face angle.• The sum of the face angles of any convex

polyhedral angle is less than .

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• The projection of a straight line upon a plane, not perpendicular to the line, is also a straight line.

• The angle that the line makes with its projection on a plane is called the angle of inclination of a line to a plane.

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Example 1, p78• is a rectangle, with and . is drawn

perpendicular to both and at . If , find the length of .

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• A solid is any limited portion of space bounded by surfaces or plane figures.

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Volume and Surface Area of Solids• The volume of a solid is the amount of space it

occupies. It has units of cubic length (i.e., cm3, m3, in3, ft3, etc.).

• The surface area is the area of a three-dimensional surface.

• The lateral area of a solid considers only the areas of the lateral or the side surfaces.

• The total surface area includes both the lateral area and the area of the bases (top and bottom). Thus, the total surface area may be defined as the total area of all surfaces that bound the solid.

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• The Cavalieri’s Principle• Given any two solids included between

parallel horizontal planes; if every right section has the same area in both solids, then the volume of the solids are equal.

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• The Volume Addition Theorem• The volume of the region enclosed by a

solid may be divided into non-overlapping smaller regions so that the sum of the volumes of these smaller regions is equal to the volume of the solid.

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• A polyhedron (plural polyhedra or polyhedrons) is a solid which is bounded by polygons joined at their edges.

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• Polyhedrons are called regular polyhedra or platonic solids if their faces are congruent regular polygons and their polyhedral angles are equal.

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• Similar Figures• Two polyhedra are said to be similar if they

have the same number of faces that are similarly placed, and which corresponding polyhedral angles are congruent. Corresponding dimensions (lengths of lines such as edge, height, etc) of similar figures are also proportional.

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Similar Figures

2

2

1

1

yx

yx

2

2

1

2

1

xx

AA

3

2

1

2

1

xx

VV

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Facts About Regular Polyhedrons

• Regular Polyhedrons of the same number of faces are similar.

• Number of edges: .where the number of polygons enclosing the polyhedron and the number of sides in each polygon.Number of vertices:

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Total Area: Volume of a Regular PolyhedronIn any regular polyhedron, where denotes the dihedral angle between any two adjacent faces, the number of faces at one vertex, and the number of sides in each polygon,

.,where denotes the number of polygons, and the length of an edge.

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Example 7

• Find the dihedral angle formed by any two adjacent faces, the total area and the volume of a regular tetrahedron if the measure of one edge is 10 inches.

• ANS: TSA=173.2 sq.in.• V=117.85 cu.in.

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Problems• #7 Page 84: The sides of an equilateral triangle

are 6 cm each. Find the distance between the plane of the triangle and a point P which is 13 cm from each vertex of the triangle.

• #9 Page 92: Find the volume of a regular dodecahedron if the total area is 2498 ft2.

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Homework

• #3, 9 Pages 83-84• #3, 5, 7 & 11 Page 92

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Lesson 4: PRISMS AND CYLINDERSSolids for which V=Bh

Week 6Math 13

Solid Mensuration

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• A prism is defined as a polyhedron with two congruent bases that lie in parallel planes, and whose every section that is parallel to a base has the same area as that of the base.

Solid Mensuration by Earnhart

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• A right prism is a prism whose lateral faces or lateral edges are perpendicular to the two bases.

• A regular prism is a right prism whose bases are regular polygons. If the base is a regular polygon of n sides then the prism contains n number of congruent lateral faces which are rectangles.

• An oblique prism is a prism whose lateral faces or lateral edges are not perpendicular to its bases. Its lateral faces are parallelograms.

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• A right section of a prism is a section made by a plane perpendicular to one of the lateral edges.

• An oblique section is made by a plane oblique to one of the lateral edges.

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Surface Areas

• The lateral area of a prism is the product of the perimeter P of a right section and the length e of a lateral edge.

Total Surface Area:

where B is the area of one base.

PeLSA

LSAB2TSA

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Volume of Prism

V = BhReBhV

sinBR

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• A rectangular solid, also known as rectangular parallelepiped is a polyhedron with two rectangular bases and lateral edges that are perpendicular to the bases.

h

wl

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Rectangular Solids

• Diagonal :

• Surface Area: .

• Volume: ..

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• A cube is a hexahedron whose 12 edges are all congruent.

d

ss

s

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Cube

• Diagonal:

• Surface Area:

• Volume: .

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• A cylinder is the solid bounded by a closed cylindrical surface and two parallel planes cutting all the elements of the surface.

• A circular cylinder is one whose bases are circles. It may also be thought of as a prism with two equal circular bases.

• A circular cylinder is a right circular cylinder, if the height or the line segment drawn through the center of the bottom base connects the center of the top base. Otherwise, the cylinder is said to be oblique.

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Surface Areas

rh2LSA LSAB2TSA

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Solid Mensuration by Earnhart

Volume of Cylinder

hrV 2

ReBhV

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Sample ProblemsEXAMPLE 2 Page 99: The trough in the figure has trapezoidal ends which lie in parallel planes. The top of the trough is a horizontal rectangle 6 ft by 16 ft and the depth of the trough is 4 ft.

– How many cubic feet of water can it hold?– How many cubic feet of water does it contain

when the depth of the water is 3 ft? – What is the area covered by water (wet portion of

the container) with this height?

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Solid Mensuration by Earnhart

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Solid Mensuration by Earnhart

EXAMPLE: The right section of a prism is in the form of a regular hexagon whose apothem measures 5 cm. If the lateral area is 360cm2, what is the length of the lateral edge of the prism?• ANS. 10.39 cm.

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EXAMPLE 10: A cylinder with a volume of 576pi m3 is circumscribed about a square prism which has one side of the base that measures 8m. What is the altitude of the cylinder?ANS: 18 m

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Solid Mensuration by Earnhart

EXAMPLE: The length of a rectangular solid is three times the width and the height is twice the width. Find the volume and the length of its diagonal if the total surface area is 198 in2. ANS: V=162 cu.in, d=11.22 in

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Solid Mensuration by Earnhart

• EXAMPLE: Find the volume of a regular hexahedron if one of the diagonals of its faces is inches.

• EXAMPLE: A cylindrical gas bottle has internal dimensions of 18 cm in diameter and 49 cm in height. It is designed to contain compressed oxygen gas. The bottle has a mass of 1.75 kg when empty and 3.15 kg when full of oxygen gas. What is the density of the oxygen gas in a full bottle in kg/m3?

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PROBLEMS1. A cylindrical tank has a lateral surface area of

88 π cm2 (not 8 π as what printed in the book) and a volume of 176π cm3. Find the base area. (#15 4.3 page 118)

2. What dimensions of a tin can of volume 54π cm3 should be produced if it is required that its height be equal to the diameter of its base?

Solid Mensuration by Earnhart

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Solid Mensuration by Earnhart

PROBLEMS

3. One edge of an oblique prism forms an angle of 300 with its projection on the plane of one base. If the lateral edge is 15 cm long and the base area is 20 cm2, find the volume of the prism.4. Each base of a right prism is a rhombus. The diagonals of a base are 12 and 6, and the altitude of the prism is 18. Find the volume and the lateral area.

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Homework

• EXERCISE 4.1: #1, 9, 17 pp. 102-104

• EXERCISE 4.2: #7, 23 pp. 110-111

• EXERCISES 4.3: #1-a, 13, 25, pp. 118-121

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Lesson 5 PYRAMIDS AND CONESSolids for which V=1/3 Bh

Week 7Math 13

Solid Mensuration

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PYRAMID• A pyramid is a polyhedron containing

triangular lateral faces with a common vertex and a base which is a polygon.

• A pyramid is a right pyramid if the line joining the vertex and the center of base is perpendicular to the plane of the base. Otherwise, the pyramid is said to be oblique.

• If the base of a right pyramid is a regular polygon then the solid is said to be a regular pyramid.

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• A pyramid is named by the type of polygon in its base.

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• The lateral surface area of a regular pyramid is equal to half the product of the perimeter of its base and its slant height.

• TOTAL SURFACE AREA:

LSABTSA

P21

LSA

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Volume of Pyramid

Where B=base area (polygon)

Bh31

V

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Solid Mensuration by Earnhart

Relationships Among Altitude, Slant Height, Lateral Edge and the Base

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CONES• A circular cone is a cone whose base is a circle. • The axis of a circular cone is the line segment joining the apex and the center of the base.• If the axis is perpendicular to the base, the cone is a right circular cone .

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• A cone whose axis is not perpendicular to the base is called oblique.

• Unless otherwise specified, a cone generally has a circular base.

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Surface Area of Cone

• Lateral Area :

• Total Surface Area:

Solid Mensuration by Earnhart

rLSA

rrTSA 2

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Solid Mensuration by Earnhart

Volume of Cone

hr31

V 2

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Solid Mensuration by Earnhart

EXAMPLE: A regular hexagonal pyramid has one lateral edge which measures 10 cm. The length of one side of its base is 6 cm. Find the altitude and the slant height of the given pyramid. Also find the volume and the total surface area. ANS: l=9.54 cm, h=8 cm, V=249.6 cu.cm.

LSA= 265.3 sq.cm.

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Solid Mensuration by Earnhart

EXAMPLE: An inverted square pyramid has a height equal to 8m and a top edge equal to 3m. Initially, it contains water to a depth of 5m. • What is the initial volume of the water in the

tank?• If additional water is to be pumped into the

tank at the rate of 20 gallons per minute, how many hours will it take to fill the tank?

• ANS: 5.86 cu.m, 4hrs

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Solid Mensuration by Earnhart

EXAMPLE: A right circular cone has a lateral area of ft2 and a base area of 512π ft2. Find the volume.

EXAMPLE: Water is flowing out of a conical funnel through its apex at a rate of 12 cubic inches per minute. If the funnel is initially full, how long will it take for it to be one-third-full? What is the height of the water level? Assume the radius to be 6 inches and the altitude of the cone to be 15 inches.

280,1

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Homework

• Exercise 5.1 pp. 131-132#’s 1-b, 1-c, 7, 17

Exercise 5.2 pp. 139-141#’s 1, 3, 23

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Lesson 6 Frustums, Truncated Solids & Prismatoids Solids for which V=(MeanB)h

Week 8Math 13

Solid Mensuration

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Solid Mensuration by Earnhart

• The frustum of a right circular cone is a portion of a right circular cone enclosed by the base of the cone, a section that is parallel to the base of the cone and the conical surface included between the base of the cone and the parallel section.

h

r1

r2

l

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Solid Mensuration by Earnhart

Relationship Among the Parts of the Frustum of a Cone

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Formulas

where and are the circumferences of its bases and is the slant height.

• where , are the two bases and is the altitude of the frustum

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Solid Mensuration by Earnhart

• The frustum of a pyramid is the lower portion of a pyramid obtained by passing a cutting plane parallel to the base intersecting all the lateral edges. Thus, it is a polyhedron enclosed by the pyramidal surface, the base of the pyramid and the parallel plane.

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Surface Areas

• .•

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Solid Mensuration by Earnhart

Volume of Frustum of a Pyramid

𝑉=13h (𝐵1+𝐵2+√𝐵1 𝐵2 )

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EXAMPLE 1

• The diameter of the lower base of a frustum of a right circular cone is while the diameter of the upper base is If the slant height of the frustum is , find the total area and the volume of the frustum.

• ANS: 1382 sq.ft, 3481 cu.ft

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EXAMPLE 2• Find the volume and the total area of a

frustum of a regular hexagonal pyramid with base edges of and , respectively, and whose altitude is

• ANS: • 769 sq.cm.• 1538 cu.cm

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Solid Mensuration by Earnhart

• A truncated circular cylinder, also known as cylindrical segment is the solid formed by passing a cutting plane through a circular cylinder intersecting all its elements.

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Solid Mensuration by Earnhart

Volume of a Truncated Cylinder

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Solid Mensuration by Earnhart

• A truncated prism is a polyhedron which is a portion of a prism cut off by a plane not parallel to the base and intersecting all the lateral edges.

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Solid Mensuration by Earnhart

• where , and is the number of sides in its base.

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Solid Mensuration by Earnhart

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Solid Mensuration by Earnhart

• A prismatoid is a polyhedron having two bases which are polygons lying in parallel planes, and lateral faces which are triangles and quadrilaterals with one side common with one base, and the opposite vertex or side common with the other base.

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• if is the altitude, is the mid-section, and and are the two base areas, respectively then the volume is

.

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Solid Mensuration by Earnhart

• A cylindrical wedge is the solid formed by passing two cutting planes through a right circular cylinder, one plane perpendicular to the axis of the cylinder and the other inclined plane intersecting the first plane through a diameter of the base.

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Solid Mensuration by Earnhart

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Solid Mensuration by Earnhart

EXAMPLE 4

• A truncated right prism has an equilateral triangular base with one side that measures . The lateral edges have lengths of and, respectively. Find the total area and the volume of the solid.

• ANS: 62.56 sq.cm, 23.4 cu.cm.

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EXAMPLE 7• The crystalline solid shown in the figure has

two parallel planes; plane is a right triangle and plane which is a

rectangle. All face angles at , and are . Find the volume of the solid.ANS: 396.55 cu.m.

Solid Mensuration by Earnhart

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Solid Mensuration by Earnhart

EXERCISES

• The volume of a frustum of a right circular cone is . The altitude is and the lower radius is three times the measure of the upper radius. Find the lateral area.

• • Find the volume of a frustum of a regular

square pyramid if the base edges are and and the measure of one lateral edge is .

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Solid Mensuration by Earnhart

• In a truncated right square prism, the two adjacent lateral edges are each long and the other two lateral edges are each long. Find the volume and the total surface area of the solid if the upper base makes an angle of with the horizontal.

• In a truncated right circular cylinder, the elliptical plane makes an angle of with the horizontal and the shortest and longest elements are and , respectively. Find the volume of the solid.

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• Find the radius of a cylindrical wedge whose volume is cubic units and whose inclined plane makes an angle of with respect to the semi-circular plane.

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Solid Mensuration by Earnhart

• Find the volume of the solid shown. All face angles at are , the lower base is rectangle and the upper base is a right triangle. All dimensions are in cm.

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Homework

• 6.1 : #’s 7, 13, 15 pp. 150-152• 6.2 : #’s 1, 5, 13, 21 pp. 159-162