Mensuration: Perimeter and Area
1. What is Mensuration?
Mensuration is the branch of mathematics that deals with measuring lengths, areas, and volumes of geometric figures.
- Perimeter: The distance around a closed figure (measured in units of length).
- Area: The amount of surface enclosed by a closed figure (measured in square units).
2. Perimeter
Perimeter of a Square
All four sides are equal.
Perimeter = 4 x side
Worked Example: Find the perimeter of a square of side 12.5 cm.
P = 4 x 12.5 = 50 cm.
Worked Example: If the perimeter of a square is 36 cm, find its side.
Side = Perimeter / 4 = 36 / 4 = 9 cm.
Perimeter of a Rectangle
Opposite sides are equal.
Perimeter = 2(length + breadth)
Worked Example: Find the perimeter of a rectangle with length 15 cm and breadth 8 cm.
P = 2(15 + 8) = 2 x 23 = 46 cm.
Common Mistake: Writing perimeter as length + breadth. That is the semi-perimeter. Always multiply by 2.
Exam Focus (3 marks): 'The perimeter of a rectangle is 60 cm. Its length is 18 cm. Find its breadth.'
2(18 + b) = 60 => 18 + b = 30 => b = 12 cm.
Perimeter of a Triangle
Perimeter = sum of all three sides
Equilateral triangle: All sides equal. P = 3 x side.
Isosceles triangle: Two sides equal. P = 2a + b (where a is the equal side, b is the base).
Scalene triangle: P = a + b + c.
Worked Example: An isosceles triangle has equal sides of 8 cm each and base 6 cm. Find its perimeter.
P = 2(8) + 6 = 16 + 6 = 22 cm.
3. Area
Area of a Square
Area = side x side = side^2
Worked Example: Find the area of a square with side 8 cm.
Area = 8 x 8 = 64 cm^2.
Worked Example: The area of a square is 121 cm^2. Find its side.
Side = square root of 121 = 11 cm.
Area of a Rectangle
Area = length x breadth
Worked Example: Find the area of a rectangle with length 14 cm and breadth 9 cm.
Area = 14 x 9 = 126 cm^2.
Worked Example: The area of a rectangle is 240 cm^2. Its length is 20 cm. Find its breadth.
Breadth = Area / Length = 240 / 20 = 12 cm.
Common Mistake: Confusing length with breadth. Both are needed, and they must be in the same unit. If length is in meters and breadth in centimeters, convert first.
Units
| Unit | Used for |
|---|---|
| mm | Very small lengths |
| cm | Everyday lengths |
| m | Larger lengths |
| km | Very large distances |
| cm^2 | Small areas |
| m^2 | Room/plot areas |
| km^2 | Large land areas |
Exam Focus (4 marks): 'A rectangular garden is 30 m long and 20 m wide. Find: (a) its perimeter (b) its area (c) the cost of fencing at 15 per meter.'
(a) P = 2(30 + 20) = 100 m.
(b) Area = 30 x 20 = 600 m^2.
(c) Cost = 15 x 100 = 1500.
Common Mistake: Using area to find the cost of fencing. Fencing goes around the boundary, so use PERIMETER, not area.
4. Word Problems
Worked Example: A rectangular piece of land measures 25 m by 15 m. A square of side 8 m is cut from it. Find the remaining area.
Area of land = 25 x 15 = 375 m^2.
Area of square cut = 8 x 8 = 64 m^2.
Remaining area = 375 - 64 = 311 m^2.
Worked Example: A wire of length 96 cm is bent into a square. Find the area of the square.
Perimeter of square = 96 cm. Side = 96 / 4 = 24 cm.
Area = 24 x 24 = 576 cm^2.
5. Comparison Table: Perimeter vs Area
| Feature | Perimeter | Area |
|---|---|---|
| Meaning | Distance around | Surface enclosed |
| Unit | m, cm, km | m^2, cm^2, km^2 |
| Square formula | 4 x side | side^2 |
| Rectangle formula | 2(l + b) | l x b |
| Application | Fencing, border | Carpeting, painting |
6. Self-Test
- Find the perimeter of a square of side 9.5 cm.
- A rectangle has length 24 cm and breadth 16 cm. Find its perimeter and area.
- The perimeter of a rectangle is 80 m. Its breadth is 15 m. Find its length and area.
- Find the area of a square whose perimeter is 48 cm.
- A rectangular field is 75 m long and 45 m wide. Find the cost of fencing at 12 per meter.
- A room is 10 m long and 8 m wide. How many square tiles of side 0.5 m are needed to tile the floor?
- Find the perimeter of an equilateral triangle of side 7.2 cm.
7. Answers to Self-Test
- P = 4 x 9.5 = 38 cm.
- P = 2(24 + 16) = 80 cm. Area = 24 x 16 = 384 cm^2.
- 2(l + 15) = 80 => l + 15 = 40 => l = 25 m. Area = 25 x 15 = 375 m^2.
- Side = 48 / 4 = 12 cm. Area = 12 x 12 = 144 cm^2.
- P = 2(75 + 45) = 240 m. Cost = 240 x 12 = 2880.
- Area = 10 x 8 = 80 m^2. Tile area = 0.5 x 0.5 = 0.25 m^2. Tiles needed = 80 / 0.25 = 320.
- P = 3 x 7.2 = 21.6 cm.
