Mensuration is the branch of mathematics dealing with the measurement of geometric figures — specifically their perimeter and area.
Perimeter
The perimeter of a closed figure is the total length of its boundary (the distance around it). It is measured in units of length (cm, m, km).
- Key perimeter formulas:
- Rectangle: P = 2 x (length + breadth) = 2(l + b)
- Square: P = 4 x side = 4s
- Triangle: P = sum of all three sides = a + b + c
- Regular polygon: P = n x side (n = number of sides)
Area
The area of a closed figure is the amount of surface enclosed within its boundary. Area is measured in square units (cm2, m2).
- Key area formulas:
- Rectangle: Area = length x breadth = l x b
- Square: Area = side x side = s2
Find the perimeter of a rectangle with length 12 cm and breadth 7 cm.
P = 2(l + b) = 2(12 + 7) = 2 x 19 = 38 cm.
Find the area of a rectangle with length 15 cm and breadth 9 cm.
Area = l x b = 15 x 9 = 135 cm2.
A square garden has perimeter 96 m. What is the area of the garden?
Side = 96/4 = 24 m. Area = 24 x 24 = 576 m2.
A triangle has sides 8 cm, 11 cm, and 13 cm. Find its perimeter.
P = 8 + 11 + 13 = 32 cm.
A rectangular room is 8 m long and 6 m wide. How many square tiles of side 0.5 m are needed to tile the floor?
Area of room = 8 x 6 = 48 m2. Area of one tile = 0.5 x 0.5 = 0.25 m2.
Number of tiles = 48 / 0.25 = 192 tiles.
A rectangular field of length 120 m and breadth 80 m needs fencing. What is the cost at Rs 15 per metre?
Perimeter = 2(120 + 80) = 400 m. Cost = 400 x 15 = Rs 6,000.
Find the area of a square whose perimeter is 28 cm.
Side = 28/4 = 7 cm. Area = 72 = 49 cm2.
Relationship between area and perimeter
Two shapes can have the same perimeter but different areas, or the same area but different perimeters. For a given perimeter, a square encloses the maximum area among rectangles.
Common mistakes
- Perimeter is the boundary (1D, measured in cm), area is the surface (2D, measured in cm2).
- A square is a rectangle, but not every rectangle is a square.
- When working with tiles, divide total area by area of one tile — do not use perimeter.
Summary
Perimeter = boundary length; area = surface enclosed. Rectangle: P=2(l+b), Area=lxb. Square: P=4s, Area=s2. These are essential tools for real-life problems involving land, flooring, fencing, and more.