DEBENDU BARMAN

VII-B2

Visual and solids shapes

With videos

mathematics

DIFFERENT TYPES OF TRIANGLE All triangles have 3 sides and 3 angles which

always add up to 180°. The Triangle Inequality Theorem states that:

  The longest side of any triangle must be   less than the sum of the other 2 sides.

Triangles are classified in 2 ways- 1) By the number of equal sides they have:

  • scalene - all 3 sides have different lengths  • isosceles - 2 sides have equal lengths  • equilateral - all 3 sides are equal

2) By the types of angles they have:  • acute triangle - all 3 angles are acute (less than 90°)  • right triangle - has one right angle (a right angle = 90°)  • obtuse triangle - has one obtuse angle (an obtuse angle is greater than 90° and less than 180°).

When these 2 categories are combined, there are 7 possible triangles:  • acute scalene (diagram A)  • right scalene (B) - all right triangles are scalene (except diagram E).  • obtuse scalene (C)

• acute isosceles (diagram D)  • right isosceles (E) also known as a 45° 45° 90° triangle.  • obtuse isosceles (F)

 • equilateral (G) all sides are equal and each angle = 60°, making this the only equiangular triangle. Since all 3 angles are less than 90° all equilateral triangles are acute triangles.

DEFFERENT TYPES OF QUADRILATERAL

his article is about four-sided mathematical shapes. For other uses, see Quadrilateral (disambiguation).

QuadrilateralSix different types of quadrilaterals

Edges and vertices4Schläfli symbol{4} (for square)Areavarious methods;see belowInternal angle(degrees)90° (for square and rectangle)In Euclidean plane geometry, a quadrilateral is a polygon with four sides (or edges) and four vertices or corners. Sometimes, the term quadrangleis used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5-sided), hexagon (6-sided) and so on.

The origin of the word "quadrilateral" is the two Latin words quadri, a variant of four, and latus, meaning "side."

Quadrilaterals are simple (not self-intersecting) or complex (self-intersecting), also called crossed. Simple quadrilaterals are either convex or concave.

The interior angles of a simple (and planar) quadrilateral ABCD add up to 360 degrees of arc, that is

This is a special case of the n-gon interior angle sum formula (n − 2) × 180°. In a crossed quadrilateral, the four interior angles on either side of the crossing add up to 720°.[1]

All convex quadrilaterals tile the plane by repeated rotation around the midpoints of their edges.

Convex quadrilaterals – parallelograms[edit]

Euler diagram of some types of quadrilaterals. (UK) denotes British English and (US) denotes American English.

A parallelogram is a quadrilateral with two pairs of parallel sides. Equivalent conditions are that opposite sides are of equal length; that opposite angles are equal; or that the diagonals bisect each other. Parallelograms also include the square, rectangle, rhombus and rhomboid.

Rhombus or rhomb: all four sides are of equal length. An equivalent condition is that the diagonals perpendicularly bisect each other. An informal description is "a pushed-over square" (including a square).

Rhomboid: a parallelogram in which adjacent sides are of unequal lengths and angles are oblique (not right angles). Informally: "a pushed-over rectangle with no right angles."[2]

Rectangle: all four angles are right angles. An equivalent condition is that the diagonals bisect each other and are equal in length. Informally: "a box or oblong" (including a square).

Square (regular quadrilateral): all four sides are of equal length (equilateral), and all four angles are right angles. An equivalent condition is that opposite sides are parallel (a square is a parallelogram), that the diagonals perpendicularly bisect each other, and are of equal length. A quadrilateral is a square if and only if it is both a rhombus and a rectangle (four equal sides and four equal angles).

.

Oblong: a term sometimes used to denote a rectangle which has unequal adjacent sides (i.e. a rectangle that is not a square).[3]

Convex quadrilaterals – other[edit]Kite: two pairs of adjacent sides are of equal length. This implies that one diagonal divides the kite into congruent triangles, and so the angles between the two pairs of equal sides are equal in measure. It also implies that the diagonals are perpendicular.Right kite: a kite with two opposite right angles.Trapezoid (North American English) or Trapezium (British English): at least one pair of opposite sides are parallel.Trapezium (NAm.): no sides are parallel. (In British English this would be called an irregular quadrilateral, and was once called a trapezoid.)Isosceles trapezoid (NAm.) or isosceles trapezium (Brit.): one pair of opposite sides are parallel and the base angles are equal in measure. Alternative definitions are a quadrilateral with an axis of symmetry bisecting one pair of opposite sides, or a trapezoid with diagonals of equal length.Tangential trapezoid: a trapezoid where the four sides are tangents to an inscribed circle

Tangential quadrilateral: the four sides are tangents to an inscribed circle. A convex quadrilateral is tangential if and only if opposite sides have equal sums.

Cyclic quadrilateral: the four vertices lie on a circumscribed circle. A convex quadrilateral is cyclic if and only if opposite angles sum to 180°.

Bicentric quadrilateral: it is both tangential and cyclic. Orthodiagonal quadrilateral: the diagonals cross at right angles. Equidiagonal quadrilateral: the diagonals are of equal length. Ex-tangential quadrilateral : the four extensions of the sides are

tangent to an excircle. More quadrilaterals[edit] An equilic quadrilateral has two opposite equal sides that,

when extended, meet at 60°. A Watt quadrilateral is a quadrilateral with a pair of opposite

sides of equal length.[4]

A quadric quadrilateral is a convex quadrilateral whose four vertices all lie on the perimeter of a square.[5]

A geometric chevron (dart or arrowhead) is a concave quadrilateral with bilateral symmetry like a kite, but one interior angle is reflex.

A self-intersecting quadrilateral is called variously a cross-quadrilateral, crossed quadrilateral, butterfly quadrilateral or bow-tie quadrilateral. A special case of crossed quadrilaterals are the antiparallelograms, crossed quadrilaterals in which (like a parallelogram) each pair of nonadjacent sides has equal length. The diagonals of a crossed or concave quadrilateral do not intersect inside the shape.

A non-planar quadrilateral is called a skew quadrilateral. Formulas to compute its dihedral angles from the edge lengths and the angle between two adjacent edges were derived for work on the properties of molecules such as cyclobutane that contain a "puckered" ring of four atoms.[6] See skew polygon for more. Historically the term gauche quadrilateral was also used to mean a skew quadrilateral.[7] A skew quadrilateral together with its diagonals form a (possibly non-regular) tetrahedron, and conversely every skew quadrilateral comes from a tetrahedron where a pair of opposite edges is removed.

Special line segments[edit] The two diagonals of a convex quadrilateral are the line segments that connect

opposite vertices. The two bimedians of a convex quadrilateral are the line segments that connect the

midpoints of opposite sides.[8] They intersect at the "vertex centroid" of the quadrilateral (see Remarkable points below).

The four maltitudes of a convex quadrilateral are the perpendiculars to a side through the midpoint of the opposite side .[9] .

OTHER

Base: the bottom surface of a solid object. Edge: the intersection of two faces on a solid

object. This is a line. Face: a flat side of a 3‐dimensional object. Prism: a solid object with two congruent and

parallel faces. Pyramid: a solid object with a polygon for a

base and triangles for sides.

CIRCLE

A circle is a simple shape of Euclidean geometry that is the set of all points in a plane that are at a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius. It can also be defined as the locus of a point equidistant from a fixed point.

A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk.

A circle can be defined as the curve traced out by a point that moves so that its distance from a given point is constant.

A circle may also be defined as a special ellipse in which the two foci are coincident and the eccentricity is 0.

Terminology[edit] Arc: any connected part of the circle. Centre: the point equidistant from the points on the circle. Chord: a line segment whose endpoints lie on the circle. Circular sector: a region bounded by two radii and an arc lying between the

radii. Circular segment: a region, not containing the centre, bounded by a chord and

an arc lying between the chord's endpoints. Circumference: the length of one circuit along the circle. Diameter: a line segment whose endpoints lie on the circle and which passes

through the centre; or the length of such a line segment, which is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord, and it is twice the radius.

Passant: a coplanar straight line that does not touch the circle. Radius: a line segment joining the centre of the circle to any point on the

circle itself; or the length of such a segment, which is half a diameter. Secant: an extended chord, a coplanar straight line cutting the circle at two

points. Semicircle: a region bounded by a diameter and an arc lying between the

diameter's endpoints. It is a special case of a circular segment, namely the largest one.

Tangent: a coplanar straight line that touches the circle at a single point.

LENGTH OF CIRCUMFERENCE[EDIT]

Further information: Circumference The ratio of a circle's circumference to

its diameter is π (pi), an irrational constant approximately equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by:

Area enclosed[edit] Area enclosed by a circle = π × area of the shaded

square Main article: Area of a disk As proved by Archimedes, the area enclosed by a

circleis equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius,[6] which comes to π multiplied by the radius squared:

Equivalently, denoting diameter by d

that is, approximately 79 percent of the circumscribingsquare (whose side is of length d).

The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely 

the isoperimetric inequality

Equations[edit] Cartesian coordinates[edit] Circle of radius r = 1, centre (a, b) = (1.2, −0.5) In an x–y Cartesian coordinate system, the circle with centre coordinates (a, b) and radius r is

the set of all points (x, y) such that

This equation, also known as Equation of the Circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram to the right, the radius is the hypotenuse of a right-angled triangle whose other sides are of length x − a and y − b. If the circle is centred at the origin (0, 0), then the equation simplifies to

The equation can be written in parametric form using the trigonometric functions sine and cosine as

where t is a parametric variable in the range 0 to 2π, interpreted geometrically as the angle that the ray from (a, b) to (x, y) makes with the x-axis. An alternative parametrisation of the circle is:

In this parametrisation, the ratio of t to r can be interpreted geometrically as the stereographic projection of the circle onto the line passing through the centre parallel to the x-axis.

In homogeneous coordinates each conic section with equation of a circle is of the form

It can be proven that a conic section is a circle exactly when it contains (when extended to the complex projective plane) the points I(1: i: 0) and J(1: −i: 0). These points are called the circular points at infinity.

Polar coordinates[edit] In polar coordinates the equation of a circle is:

where a is the radius of the circle,  is the polar coordinate of a generic point on the circle, and  is the polar coordinate of the centre of the circle (i.e., r0 is the distance from the origin to the centre of the circle, and φ is the anticlockwise angle from the positive x-axis to the line connecting the origin to the centre of the circle). For a circle centred at the origin, i.e. r0 = 0, this reduces to simply r = a. When r0 = a, or when the origin lies

on the circle, the equation becomes

In the general case, the equation can be solved for r, giving

the solution with a minus sign in front of the square root giving the same curve. Complex plane[edit] In the complex plane, a circle with a centre at c and radius (r) has the

equation . In parametric form this can be written .

The slightly generalised equation   for real p, q and complex g is sometimes called a generalised circle. This becomes the above equation for a circle with  since . Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a line.

Tangent lines[edit] Main article: Tangent lines to circles The tangent line through a point P on the circle is perpendicular to the

diameter passing through P. If P = (x1, y1) and the circle has centre (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x1, y1), so it has the form (x1 − a)x + (y1 – b)y = c. Evaluating at (x1, y1) determines the value of c and the result is that the equation of the tangent is

Or

If y1 ≠ b then the slope of this line is

This can also be found using implicit differentiation.

When the centre of the circle is at the origin then the equation of the tangent line becomes

and its slope is

CONE A cone is a three-dimensional geometric shape that tapers

smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.

More precisely, it is the solid figure bounded by a base in a plane and by a surface (called the lateral surface) formed by the locus of all straight line segments joining the apex to the perimeter of the base. The term "cone" sometimes refers just to the surface of this solid figure, or just to the lateral surface.

The axis of a cone is the straight line (if any), passing through the apex, about which the base has a rotational symmetry.

In common usage in elementary geometry, cones are assumed to be right circular, where circular means that the base is a circle and rightmeans that the axis passes through the centre of the base at right angles to its plane. Contrasted with right cones are oblique cones, in which the axis does not pass perpendicularly through the centre of the base.[1] In general, however, the base may be any shape and the apex may lie anywhere (though it is usually assumed that the base is bounded and therefore has finite area, and that the apex lies outside the plane of the base).

CYLINDER A cylinder (from Greek κύλινδρος – kulindros, "roller,

tumbler"[1]) is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder. The surface area and the volume of a cylinder have been known since deep antiquity.

In differential geometry, a cylinder is defined more broadly as any ruled surface spanned by a one-parameter family of parallel lines. A cylinder whosecross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder respectively.

The open cylinder is topologically equivalent to both the open annulus and the punctured plane.

SPHERE A sphere (from Greek σφαῖρα — sphaira, "globe, ball"[1])

is a perfectly round geometrical and circular object in three-dimensional space that resembles the shape of a completely round ball. Like a circle, which, in geometrical contexts, is in two dimensions, a sphere is defined mathematically as the set of points that are all the same distance r from a given point in three-dimensional space. This distance r is the radius of the sphere, and the given point is the center of the sphere. The maximum straight distance through the sphere passes through the center and is thus twice the radius; it is the diameter.

In mathematics, a distinction is made between the sphere (a two-dimensional closed surface embedded in three-dimensional Euclidean space) and theball (a three-dimensional shape that includes the interior of a sphere).