MENSURATION
In this lesson we will cover:
Basic methods of derivating perimeters of 2d
objects [PAGES 1-4]
Basic methods of derivating areas of 2d
objects [PAGES 5-7]
Basic methods of derivating volumes of 3d
objects [PAGES 8-11]
PAGE 1
PERIMETER
Perimeter can be defined as the length of the boundary of a 2
dimensional shape.
4 cm
2 cm 2 cm
4 cm
For example, the length of the sides of the given rectangle are 4cm, 2cm,
4cm, and 2cm. The combined length of all the sides gives us the
perimeter; i.e. 4+2+4+2=12 cm.
PAGE 2
PERIMETER
Similary, perimeter can be followed in a similar way (by adding all the
sides of the shape) as long as the shape does not have any curves.
x cm
Rectangle: Let the sides be x and y cm.
P = x+y+x+y cm y cm y cm
P = 2x+2y cm
P = 2(x+y) cm x cm
(Opposite sides are equal in a rectangle)
x cm
Square: Let the sides be x cm.
P = x+x+x+x cm
x cm x cm
P = 4x cm
(All sides are equal in a square)
x cm
PAGE 3
PERIMETER
Triangle: Let the sides be x, y, and z cm.
P = x+y+z cm (In a scalene triangle: All three
sides are of different lengths)
P = x+x+y cm y cm z cm
P = 2x+y cm (In an isosceles triangle: Two
sides are equal)
P = x+x+x cm x cm
P = 3x cm (In an equilateral triangle: All three
sides are equal)
Note: In a regular 2d polygon (where all sides are equal), the perimeter is
equal to the length of one side * the number of sides.
Example: Let the length of a side of an octagon (8-sided polygon) be x.
P=x*8
P = 8x
A regular triangle is equilateral. A regular quadrilateral is a square, not a
rectangle.
PAGE 4
PERIMETER
In a circle, the perimeter is also known as the circumference of the circle.
Circle: Let the radius (distance from the
centre to the boundary) be r cm.
C = 2 * π * r cm
r cm
P = 2πr cm
(Take π = 3.14 or 22/7)
Similarly, perimeters can be found out for other polygons by adding the
all the sides if they are unequal, else if they are equal, multiply the length
of one side by the number of sides. For example = p of rhombus can be
found out in a way similar to the square, p of a parallelogram can be
found out in a way similar to a rectangle.
Note: A circle is NOT a polygon as it has a curved boundary.
PAGE 5
AREA
The area of a 2-dimensional figure tells us how much space they occupy
on the piece of paper. Its unit is (unit)².
5 cm
Area of rectangle = length *
breadth 2 cm
2 cm
A = 5 * 2 sq. cm
A = 10 sq. cm
5 cm
3 cm
Square = side * side
A = 3 * 3 sq. cm
3 cm 3 cm
A = 9 sq. cm
3 cm
PAGE 6
AREA
A triangle can be considered as 1/2 of a rectangle. Taking length of
rectangle as the base of the triangle, and the breadth of rectangle as the
height of the triangle, we can find out its area.
Area of triangle = 1/2 * height *
base h
h
For example: If h= 4 cm, b= 3cm,
A= 1/2 * 4 * 3 sq. cm
A= 6 sq. cm
b b
The area of an equilateral
triangle can also be written as
(√3)/4 side sq. unit.
h
b
PAGE 7
AREA
Area of circle = πr² unit
For example: If r= 7 cm,
r cm
A= π * 7² cm
A= 22/7 * 7 * 7 cm
A= 154 cm²
Area of semicircle = 1/2πr² unit
(half of a circle)
For example: If r= 7 cm, r cm
A= 1/2 * π * 7² cm
A= 1/2 * 22/7 * 7 * 7 cm
A= 77 cm²
PAGE 8
VOLUME
The volume can be defined as the space which a 3-dimensional object
occupies. As area is for 2-dimensional shapes, volume is for 3-
dimensional shapes. Its unit is (unit)³.
Since in a cube, length = width
= height = length of one edge,
Area of cube = s cm³ (s = length
of the edge)
For example, if s = 3 cm,
A = s³
A = 3 * 3 * 3 cm³
A = 27 cm³
PAGE 9
VOLUME
Volume of cuboid= length *
depth * height
For example, if l= 5 cm, d=3 cm,
h= 4cm,
A = l*d*h cm³
A = 5 * 3 * 4 cm³
A = 60 cm³
Volume of right cylinder= πr²h
For example, if r=7 cm, h= 4cm,
A = π * r² * h cm³
A = 22/7 * 7 * 7 * 4 cm³
A = 616 cm³
PAGE 10
VOLUME
Volume of right circular cone=
1/3πr²h (1/3rd of cylinder)
For example, if r=7 cm, h= 4cm,
A = 1/3 * π * r² * h cm³
A = 1/3 * 22/7 * 7 * 7 * 4 cm³
A = 205.33 cm³
Volume of sphere= 4/3πr³
For example, if r= 7 cm,
A = 4/3 * π * r³ cm³
A = 4/3 * 22/7 *7³ cm³
A = 1437.33 cm³
PAGE 11
VOLUME
Volume of hemisphere= 2/3πr³
(half of a sphere)
For example, if r= 7 cm,
A = 2/3 * π * r³ cm³
A = 2/3 * 22/7 *7³ cm³
A = 718.66 cm³
