**MENSURATION**

We will be covering the following figures

Square

2-D

A shape that only has two dimensions (such as width and height) and no thickness.

What is a

Two Dimensional Figure?

We will be covering the following figures:

Cube

Cuboid

Frustum

Cylinder

Cone

Sphere

3-D

A 3D figure is a figure that has height, depth, and width. Such a figure does not lie entirely in a plane. The "3" in 3D refers to the number of dimensions. The "D" stands for dimension, which is an axis in a coordinate system.

What is a

Three Dimensional Figure?

Square

A square is a regular quadrilateral.

This means that it has four equal sides and four equal angles (90-degree angles, or right angles).

It can also be defined as a rectangle in which two adjacent sides have equal length.

Area of a square

Perimeter of a Square

The Perimeter of a Square is as follows:

Perimeter= sum of the length of all sides

which gives you

P= 4s ( "s" being the side of the square )

Rectangle

A rectangle is any quadrilateral with four right angles.

It can also be defined as an

equiangular quadrilateral.

It can also be defined as a parallelogram containing a right angle.

Area of a

Rectangle

Much like the area of a square, the same formula applies to the rectangle

but there's a slight difference, as the rectangle does not have equal sides, the formula is:

Area = length x breadth

Perimeter of a rectangle

The perimeter is the length of the boundry of the rectangle, so the perimeter would be:

Perimeter = 2 X the Length + 2 X the Width

the diagram on the right will help you understand this a little better

A triangle is one of the basic shapes of geometry.

It's a polygon with three corners or vertices and three sides or edges which are line segments.

A triangle with vertices A, B, and C is denoted .

There are six types of triangles

Isosceles

Scalene

Equilateral

Right angle Triangle

Obstuse Angle Triangle

Acute Angle Triangle

Area of a Triangle

The area of a Triangle has a very easy formula

which is :

Area of a = 1/2 X h X b

(where "h" is height; "b" is base)

Perimeter of a Triangle

Perimeter is the boundry of the triangle so the

perimeter would be :

Perimeter = sum of all sides

Circle

A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior.

The terminologies in association with a circle are given in the box below

A circle's

Diameter

is the length of a line segment whose endpoints lie on the circle and which passes through the centre.

The

Radius

is half of the diameter

A

chord

is a line segment whose endpoints lie on the circle. (eg) the diameter is the longest chord in a circle

A

tangent

is a perpendicular line which touches any point on the circle's circumference

A

secant

is an Extended chord which runs through the circle

Trapezoid

An

arc

is a part of the circle's circumference

A

sector

is a region bounded by two radii and an arc lying between the radii

A

segment

is a region bounded by a chord and an arc lying between the chord's endpoints.

Area of a circle

The area of a circle is:

Perimeter/circumference of a circle

The circumference of the circle:

C=

Arc length

To find the length of an arc, we use this formula:

Area of Sector

Area of Arc=

A trapezoid is a quadrilateral with ONE pair of parallel sides

The parallel sides are called the bases of the trapezoid and the other two sides are called the legs or the lateral sides.

Area and perimeter of a trapezium

The area of a trapezium:

A = 1/2 X sum of parallel sides X height

Perimeter of a trapezium

P = the sum of parallel sides + length of slanting sides

Cube

Cuboid

Frustrum

Cylinder

Cone

Sphere

A cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

²

²

²

Cube is also a cuboid.

Volume of a cuboid = length x breadth x height

All the sides of a cube are equal.

Thus, Volume of a cube = a x a x a

a

a

a

A cuboid is a solid figure bounded by six faces

Opposite sides have same area in a cuboid,

which makes the surface area formula

= b x h + b x h + l x b + l x b + l x h + l x h

which can also be written as:

total surface area = 2( bh + lb + lh )

l x b

l x b

b x h

b x h

l x h

l x h

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball.

Like a circle, which is in two dimensions, a sphere is the set of points which are all the same distance r from a given point in space.

This distance r is known as the "radius" of the sphere, and the given point is known as the center of the sphere.

The maximum straight distance through the sphere is known as the "diameter".

It passes through the center and is thus twice the radius.

A cylinder 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.

Basically a cylinder is just a circle which has been streched into a third dimension.

A frustum of a cone is the slice of a right circular cone between two planes that are perpendicular to the axis of symmetry ofthe cone. Its slant height (in the diagram) is the length of asegment from the edge of thetop to the edge of the bottom, perpendicular to both. We'll denote the radii of the top and bottom circles by r and R respectively

l = length

b = breadth

h = height

The volume of a cylinder :

volume = cross-sectional area x height

Since the cross - sectional area is the area of one of the circles on one side,

The volume of a cylinder is:

2

2

A cone is an -dimensional geometric shape that tapers smoothly from a base (usually flat and circular) to a point called the apex or vertex.

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

Triangle

A Few Parts of Circles

Arc Length =

n = central angle

To find the area of a sector we use

It is normally a cone or pyramid

Conical Frustum

Volume Of Conical Frustum

Volume of a Conical Frustum is Calculated by the formula

where R1, R2 are the radii of the two bases.

In geometry, the rhombic triacontahedron is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types

Rhombic triacontahedron

Surface Area of

Rhombic triacontahedron

Surface Area of Rhombic triacontahedron is calculated by the formula

Net of

Rhombic triacontahedron

S.A =

Volume of

Rhombic triacontahedron

Volume of a of Rhombic triacontahedron can be calculated by the formula

V=

where edge length of a rhombic triacontahedron is 'a'

where edge length of a rhombic triacontahedron is 'a

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of two types.

rhombic dodecahedron

Surface Area of

rhombic dodecahedron

Surface area of rhombic dodecahedron can be calculated by the formula

S.A. =

Net of

Rhombic Dodecahedron

Volume of a

rhombic dodecahedron

Volume of Rhombic Dodecahedron can be calculated by

v=

Volume of Cylinder

The formula for finding the surface area of a cylinder is, with h as height, r as radius, and S as surface area is

Surface Area of Cylinder

where l is the lateral height of the cone

Surface area of cone can be calculated by the formula

Surface Area of Cone

Volume of Cone

The volume V of any conic solid is one third of the product of the area of the base B and the height H is

Surface area of Sphere

Surface Area of Sphere can be calculated by the formula

In 3 dimensions, the volume inside a sphere (that is, the volume of a ball) is derived to be

where r is the radius of the sphere

Volume of Sphere

Some exciting complex shapes

That was all about.....

Rectangle

Circle

Triangle

Trapezium

Area of Square can be calculated by the formula

where 'a' is the measure of the side of the square

Parallelogram

A Parallelogram is a simple (non self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.

Parallelogram

The area K of the parallelogram to the right (the dark area) is the total area of the rectangle less the area of the two light color triangles.

Area of Parallelogram

Day In And Day Out, We See A Variety of objects and Shapes All Around Us

many of these objects play an integral part in our lives

But have we ever really thought about these objects ?

do they have a known shape?

how much space might they occupy?

do these shapes have definite properties?

can these properties be studied mathematically ?

How can this study help us?

it is this very study

that we call