Mensuration is the branch of mathematics that deals with the measurement of lengths, areas, volumes and capacities of plane figures and solid shapes. We use mensuration constantly in daily life: to find the area of a field, the length of a fence around a garden, the surface area of a water tank, or the volume of a box to be painted or filled.
In this chapter we will learn how to find the area and perimeter of plane figures, including the area of a trapezium, a general quadrilateral, a polygon and a rhombus. We will also study the surface area and volume of solid figures such as cubes, cuboids and cylinders. Along the way we will learn about the important concept of the volume and capacity, and how to convert between units of area and volume.
A trapezium is a quadrilateral with one pair of parallel sides. Its area is: $$\text{Area} = \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}$$
If the parallel sides are a and b and the height is h, then area = (1/2)(a + b)h.
A general quadrilateral can be divided by a diagonal into two triangles. Its area is the sum of the areas of the two triangles. $$\text{Area} = \frac{1}{2} d(h_1 + h_2)$$ where d is the diagonal and h1, h2 are the heights of the two triangles from the opposite vertices.
A polygon can be split into triangles, trapeziums and rectangles, and the sum of their areas gives the area of the polygon. This method is called dividing the polygon into known shapes.
The area of a rhombus can be found using its diagonals: $$\text{Area} = \frac{1}{2} \times (\text{product of diagonals})$$
If the diagonals are d1 and d2, then area = (1/2) d1 d2.
A cuboid has length l, breadth b and height h. Its total surface area is: $$\text{Total Surface Area} = 2(lb + bh + hl)$$
Its lateral surface area (area of four walls, excluding top and bottom) is: $$\text{Lateral Surface Area} = 2h(l + b)$$
A cube has all edges equal to a. Its total surface area is: $$\text{Total Surface Area} = 6a^2$$
Its lateral surface area is 4a^2.
A cylinder with radius r and height h has: $$\text{Curved Surface Area} = 2\pi rh$$ $$\text{Total Surface Area} = 2\pi r(h + r)$$
Volume is the measure of the space occupied by a solid. It is measured in cubic units like cubic cm or cubic m.
$$\text{Volume} = l \times b \times h$$
$$\text{Volume} = a^3$$
$$\text{Volume} = \pi r^2 h$$
The capacity of a container is the volume of the liquid it can hold. Usually capacity is measured in millilitres (mL) or litres (L). Note that 1 L = 1000 mL and 1 cm^3 = 1 mL.
1 m^2 = 10000 cm^2 (since 1 m = 100 cm). 1 cm^2 = 100 mm^2.
1 m^3 = 1000000 cm^3 (since 1 m = 100 cm). 1 cm^3 = 1 mL, and 1000 cm^3 = 1 L.
When converting from a larger unit to a smaller unit, we multiply, and from a smaller unit to a larger unit, we divide.
| Shape | Formula |
|---|---|
| Rectangle | l x b |
| Square | side x side = a^2 |
| Triangle | (1/2) x base x height |
| Trapezium | (1/2) x (sum of parallel sides) x height |
| Rhombus | (1/2) x d1 x d2 |
| Circle | pi x r^2 |
| Solid | Total surface area | Volume |
|---|---|---|
| Cuboid | 2(lb + bh + hl) | l x b x h |
| Cube | 6a^2 | a^3 |
| Cylinder | 2 x pi x r x (r + h) | pi x r^2 x h |
Mensuration gives us the tools to measure the world around us. In this chapter we learnt how to find the areas of plane figures such as the trapezium, general quadrilaterals, polygons and rhombuses, using their formulas and the method of dividing figures into known shapes. We then moved to solid shapes and studied the surface areas and volumes of cuboids, cubes and cylinders, along with the important idea of capacity and the conversion of units of area and volume. These skills are used in architecture, engineering, agriculture, construction and countless everyday activities. Mastery of mensuration formulas and careful attention to units will help students score well and use mathematics practically throughout their lives.