Mensuration is the branch of mathematics that deals with the measurement of lengths, areas and volumes. When we measure the length of a table, the area of a floor, or the amount of water a tank can hold, we are using mensuration. These measurements are important in daily life: we need the perimeter of a field to fence it, and the area of a wall to paint it.
Perimeter is the total distance around the boundary of a closed figure. It is measured in units of length such as centimetres, metres and kilometres. Area is the amount of surface covered by a closed figure. It is measured in square units such as square centimetres and square metres. In this chapter, we learn to find the perimeter of rectangles, squares and other polygons, and the area of rectangles and squares.
We also compare figures by their areas and learn about the relationship between area and perimeter. For example, two different figures can have the same perimeter but different areas, and vice versa. This understanding is the first step towards studying volumes and surface areas of solid shapes in higher classes.
The perimeter of a closed figure is the total length of its boundary. To find the perimeter, we add the lengths of all the sides of the figure.
Perimeter of a figure = Sum of the lengths of all its sides.
For example, a field with sides 5 m, 7 m, 5 m and 7 m has a perimeter of 5 + 7 + 5 + 7 = 24 m. Perimeter is measured in units of length: millimetres, centimetres, metres or kilometres.
A rectangle has two equal lengths and two equal breadths. The perimeter of a rectangle is the sum of its four sides.
Perimeter of a rectangle = length + breadth + length + breadth = 2 x (length + breadth).
For example, a rectangle of length 10 cm and breadth 6 cm has perimeter 2 x (10 + 6) = 2 x 16 = 32 cm.
A square has four equal sides. The perimeter of a square is the sum of its four equal sides.
Perimeter of a square = side + side + side + side = 4 x side.
For example, a square of side 5 cm has perimeter 4 x 5 = 20 cm.
The perimeter of a triangle is the sum of the lengths of its three sides. For a triangle with sides a, b and c, the perimeter is a + b + c.
For example, a triangle with sides 4 cm, 5 cm and 6 cm has perimeter 4 + 5 + 6 = 15 cm.
For irregular shapes, we simply add the lengths of all the sides of the figure.
The area of a closed figure is the amount of surface enclosed by the figure. Area is measured in square units. A square unit is the area of a square whose side is 1 unit long. For example, a square of side 1 cm has an area of 1 square centimetre (1 sq cm or 1 cm squared).
The area of a rectangle is the product of its length and its breadth.
Area of a rectangle = length x breadth.
For example, a rectangle of length 10 cm and breadth 6 cm has area 10 x 6 = 60 sq cm.
Since all sides of a square are equal, the area of a square is the product of its side with itself.
Area of a square = side x side = side squared.
For example, a square of side 5 cm has area 5 x 5 = 25 sq cm.
To find the area of an irregular figure drawn on a square grid (graph paper), we count the number of squares covered by the figure. Squares that are fully covered are counted as full squares. Squares covered half or more are counted as half squares, and squares covered less than half are ignored.
Area and perimeter are different concepts. Perimeter measures the boundary (length), while area measures the surface (space enclosed). A figure can have a small perimeter but a large area, or a large perimeter but a small area.
For example, a rectangle 6 m by 2 m has perimeter 2 x (6 + 2) = 16 m and area 12 sq m. A rectangle 5 m by 3 m also has perimeter 16 m but area 15 sq m. So the same perimeter can give different areas.
Similarly, a square and a rectangle can have the same area but different perimeters. For a fixed area, the square usually has the least perimeter.
If the perimeter of a rectangle and one of its dimensions are given, we can find the other dimension.
Perimeter = 2 x (length + breadth), so length + breadth = Perimeter divided by 2. Then subtract the known dimension.
For example, if the perimeter of a rectangle is 40 m and its length is 12 m, then length + breadth = 20 m, so breadth = 20 - 12 = 8 m.
If the perimeter of a square is given, each side equals Perimeter divided by 4.
For example, a square with perimeter 36 cm has side 36 / 4 = 9 cm.
If the area of a square is given, each side equals the square root of the area.
For example, a square with area 25 sq cm has side 5 cm, since 5 x 5 = 25.
Perimeter is a length, so it uses units of length: cm, m, km. Area uses square units: sq cm, sq m, sq km.
When solving problems, we must make sure all measurements are in the same units before adding or multiplying.
Mensuration is used in many practical situations: - Finding the length of a fence needed for a garden (perimeter). - Finding the amount of carpet or tiles needed for a floor (area). - Finding the length of ribbon needed to decorate the border of a picture (perimeter). - Finding the area of a wall to be painted.
| Figure | Perimeter | Area |
|---|---|---|
| Rectangle | 2 x (length + breadth) | length x breadth |
| Square | 4 x side | side x side |
| Triangle | Sum of three sides | - |
| Unit | Conversion |
|---|---|
| 1 m | 100 cm |
| 1 km | 1000 m |
| 1 sq m | 10,000 sq cm |
| 1 sq km | 1,000,000 sq m |
Mensuration teaches us how to measure the boundaries and surfaces of figures. We learnt to find the perimeter of rectangles, squares and triangles, and the area of rectangles and squares. We also understood the difference between perimeter and area, the importance of units, and how to use these ideas in real-life situations like fencing and flooring. These foundations prepare us for studying circles, triangles and solid shapes in higher classes.