CONFIDENTIAL1 Geometry Developing Formulas for Circles and Regular Polygons.
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Transcript of CONFIDENTIAL1 Geometry Developing Formulas for Circles and Regular Polygons.
CONFIDENTIAL 1
Geometry
Developing Formulas for Circles and
Regular Polygons
CONFIDENTIAL 2
Warm up
Find the each measurement:
1) d2 of a kite if A = 14 cm2 and d1 = 20 cm.
2) the area of a trapezoid in which b1 = 3 yd, b2 = 6 yd and h = 4 yd
CONFIDENTIAL 3
A circle is the locus of a point in a plane that are a fixed distance from a point called the center of the circle. A circle is named by the symbol and its center. A has
a radius r= AB and diameter d = CD.
Formulas for Circle
The irrational number ∏ is defined as the ratio of the circumference C to the
diameter d, or ∏ = C. d
Solving for C gives the formula C= ∏d.Also d = 2r, so C= 2∏r. C D
B
A
CONFIDENTIAL 4
You can use the circumference of a circle to find its area. Divide the circle and rearrange the pieces to make a
shape that resembles a parallelogram.
The base of the parallelogram is about half the circumference, or ∏r, and the height is close to the radius r.
So A ≈ ∏r. r = ∏r2.
∏r
r
Formulas for Circle
CONFIDENTIAL 5
The more pieces you divide the circle into, the more accurate the estimate will be.
Formulas for Circle
CONFIDENTIAL 6
A circle with diameter d and radius r has circumference C= ∏d or C= 2∏r and area A = ∏r2.
Circumference and Area of a Circle
diameterra
diu
s
CONFIDENTIAL 7
Finding measurements of of Circle
16 cm
P
Find each measurement:
A = ∏r2
A = ∏(8)2
A = 64∏ cm2
Area of a circle.
Divide the diameter by 2 to find the radius, 8.
Simplify.
CONFIDENTIAL 8
Finding measurements of of CircleFind each measurement:
B) the radius of X in which C = 24∏ in.
C = 2∏r
24A = 2∏r
r = 12 in
Circumference of a circle.
Substitute 24∏ for C.
Divide both sides by 2∏.
CONFIDENTIAL 9
Finding measurements of of Circle
C) the circumference of S in which A = 9x2∏ cm2.
A = ∏r2
9x2∏ = ∏r2
9x2= 2r2
3x= r
Area of a circle.Substitute 9x2 for A.Divide both sides by 2.Take square root of both sides.
Step 1: Use the given area to solve for r.
C = 2∏rC = 2∏(3x)C = 6x∏
Substitute 3x for r.Divide both sides by 2.Simplify.
Step 2: Use the value of r to find the circumference.
CONFIDENTIAL 10
Now you try!
1) Find the area of A in terms of ∏ in which C = (4x - 6)∏ m.
CONFIDENTIAL 11
Music Application
10 in. diameterA = ∏(5)2 since, r = 10 2 A≈ 78.5 in2
14 in. diameterA = ∏(7)2 since, r = 12 2 A≈ 153.9 in2
A drum kit contains three drums with diameters of 10 in., 12 in. and 14 in. Find the area of the top of each drum.
Round to the nearest tenth.
12 in. diameterA = ∏(6)2 since, r = 10 2 A≈ 113.1 in2
CONFIDENTIAL 12
Now you try!
2) A drum kit contains three drums with diameters of 10 in., 12 in. and 14 in. Find the circumference of the
top of each drum. Round to the nearest tenth.
CONFIDENTIAL 13
Formulas for a Regular Polygon
The center of a regular polygon is equidistant from the vertices. The apothem is the distance from the center to a
side. A central angle of a regular polygon has its vertex at the center, and its side pass through consecutive vertices. Each central angle measure of a regular n - gon is 360°.
n
C
H
B
DG
F E
Regular pentagon DEFGH has center C, apothem BC,
and central angle /DCE.
CONFIDENTIAL 14
To find the area of a regular n - gon with side length s and apothem a, divide it into n congruent isosceles
triangles.
area of each triangle : 1as 2
total area of the polygon: A = n. 1as 2
or A= 1aP 2
The perimeter is P = ns
Formulas for a Regular Polygon
CONFIDENTIAL 15
The area of a regular polygon with apothem a and perimeter P is
A = 1aP 2
Formulas for a Regular Polygon
CONFIDENTIAL 16
A) a regular hexagon with side length 6 m
The perimeter is 6(6)= = 36 m .
Finding the area of a Regular PolygonFind the area of each regular polygon. Round to the nearest tenth.
Area of a regular polygon
Substitute 3√3 for a and 36 for P
Simplify
6 m
3m3√3m
A= 1aP 2
A= 1(3√3)(36) 2
A= 54√3≈ 93.5 m2
The hexagon can be divided into 6 equilateral triangles with side length 6 m. By the 30°-60°- 90° Triangle theorem, the apothem is
3√3.
CONFIDENTIAL 17
B) a regular pentagon with side length 6 in.
36°
4 in
a
Step 1: Draw the pentagon. Draw an isosceles triangle with its vertex at the centre of the pentagon. The
central angle is 360° = 72 ° . 5
Draw a segment that bisects the central angle and the side of the polygon to form a right triangle.
CONFIDENTIAL 18
36°
4 in
a
Step 2: Use the tangent ratio to find the apothem.
The tangent of an angle is opp. leg adj. leg
Solve for a.
tan 36° = 4 a a = 4 tan36°
CONFIDENTIAL 19
Step 3 : Use the apothem and the given side length to find the area .
Area of a regular polygon.
The perimeter is 8(5) = 40 in.
Simplify. Round to the nearest tenth.
A= 1aP 2 A = 1. 4 .(40) 2 tan36°
A ≈ 101.1 in2
36°
4 in
a
CONFIDENTIAL 20
Now you try!
3) Find the area of a regular octagon with a side of 4 cm.
CONFIDENTIAL 21
Now some problems for you to practice !
CONFIDENTIAL 22
Find each measurement:
Assessment
1) the circumference of C. 10 cm∏C
2) the area of A in terms of ∏.3x inA
3) the circumference of Pin which A = 36∏ ft2.
CONFIDENTIAL 23
4) a food pizza parlor offers pizzas with diameters of 8 in., 10 in. and 12 in. Find the area of each pizza size.
Round to the nearest tenth.
CONFIDENTIAL 24
5) 6)
Find the area of each regular polygon. Round to the nearest tenth:
10 in
3 cm
CONFIDENTIAL 25
7) An equilateral triangle with an apothem length of 2 ft.
8) An regular dodecagon with an side length of 5 ft.
Find the area of each regular polygon. Round to the nearest tenth:
CONFIDENTIAL 26
A circle is the locus of a point in a plane that are a fixed distance from a point called the center of the circle. A circle is named by the symbol and its center. A has
a radius r= AB and diameter d = CD.
Formulas for Circle
The irrational number ∏ is defined as the ratio of the circumference C to the
diameter d, or ∏ = C. d
Solving for C gives the formula C= ∏d.Also d = 2r, so C= 2∏r. C D
B
A
Let’s review
CONFIDENTIAL 27
A circle with diameter d and radius r has circumference C= ∏d or C= 2∏r and area A = ∏r2.
Circumference and Area of a Circle
diameterra
diu
s
CONFIDENTIAL 28
Finding measurements of of Circle
C) the circumference of S in which A = 9x2∏ cm2.
A = ∏r2
9x2∏ = ∏r2
9x2= 2r2
3x= r
Area of a circle.Substitute 9x2 for A.Divide both sides by 2.Take square root of both sides.
Step 1: Use the given area to solve for r.
C = 2∏rC = 2∏(3x)C = 6x∏
Substitute 3x for r.Divide both sides by 2.Simplify.
Step 2: Use the value of r to find the circumference.
CONFIDENTIAL 29
Music Application
10 in. diameterA = ∏(5)2 since, r = 10 2 A≈ 78.5 in2
14 in. diameterA = ∏(7)2 since, r = 12 2 A≈ 153.9 in2
A drum kit contains three drums with diameters of 10 in., 12 in. and 14 in. Find the area of the top of each drum.
Round to the nearest tenth.
12 in. diameterA = ∏(6)2 since, r = 10 2 A≈ 113.1 in2
CONFIDENTIAL 30
Formulas for a Regular Polygon
The center of a regular polygon is equidistant from the vertices. The apothem is the distance from the center to a
side. A central angle of a regular polygon has its vertex at the center, and its side pass through consecutive vertices. Each central angle measure of a regular n - gon is 360°.
n
C
H
B
DG
F E
Regular pentagon DEFGH has center C, apothem BC,
and central angle /DCE.
CONFIDENTIAL 31
The area of a regular polygon with apothem a and perimeter P is
A = 1aP 2
Formulas for a Regular Polygon
CONFIDENTIAL 32
A) a regular hexagon with side length 6 m
The perimeter is 6(6)= = 36 m .
Finding the area of a Regular PolygonFind the area of each regular polygon. Round to the nearest tenth.
Area of a regular polygon
Substitute 3√3 for a and 36 for P
Simplify
6 m
3m3√3m
A= 1aP 2
A= 1(3√3)(36) 2
A= 54√3≈ 93.5 m2
The hexagon can be divided into 6 equilateral triangles with side length 6 m. By the 30°-60°- 90° Triangle theorem, the apothem is
3√3.
CONFIDENTIAL 33
You did a You did a great great job today!job today!