Perimeter is the total distance around the boundary of a closed figure. It is measured in units of length such as centimetres, metres or kilometres. Area is the amount of surface enclosed by a closed figure; it is measured in square units such as square centimetres or square metres. Fencing a field, framing a picture, tiling a floor and laying a carpet all involve perimeter and area in practical ways. If we know the perimeter, we can compute the length of fencing needed; if we know the area, we can decide how much tile, paint or grass is required.
In this chapter we study the perimeter and area of triangles, rectangles, squares, parallelograms and circles, and also the area of composite figures made by combining these shapes. We learn to convert between units of area, such as square metres and square kilometres, and between units of volume for cubes and cuboids. Perimeter and area form the subject of mensuration, one of the most directly useful branches of mathematics for construction, agriculture, architecture and everyday problem solving.
A rectangle has length l and breadth b. Its perimeter is twice the sum of its length and breadth: Perimeter of rectangle = 2 x (l + b) For example, a rectangle with length 8 cm and breadth 5 cm has perimeter 2 x (8 + 5) = 26 cm.
A square has four equal sides of length s. Its perimeter is four times the side: Perimeter of square = 4 x s For example, a square of side 6 cm has perimeter 4 x 6 = 24 cm.
The perimeter of a triangle is the sum of its three sides. If the triangle has sides a, b and c: Perimeter of triangle = a + b + c For example, a triangle with sides 4 cm, 5 cm and 6 cm has perimeter 15 cm. In an equilateral triangle of side s, the perimeter is 3 x s.
The area of a rectangle is the product of its length and breadth: Area of rectangle = l x b For example, a rectangle with length 8 cm and breadth 5 cm has area 8 x 5 = 40 square cm.
The area of a square is the square of its side: Area of square = s x s = s^2 For example, a square of side 6 cm has area 36 square cm.
The area of a triangle is half the product of its base and height: Area of triangle = (1/2) x base x height For example, a triangle with base 10 cm and height 6 cm has area (1/2) x 10 x 6 = 30 square cm.
A parallelogram has a base b and a height h, where the height is the perpendicular distance between the base and its opposite side. Its area is: Area of parallelogram = base x height = b x h For example, a parallelogram with base 12 cm and height 5 cm has area 12 x 5 = 60 square cm. A rectangle is a special parallelogram, and its area formula is the same. A triangle formed by cutting a parallelogram along a diagonal has exactly half the area of the parallelogram.
A circle is a set of all points at a fixed distance, called the radius r, from a fixed point called the centre. The diameter d is twice the radius: d = 2 x r. The number pi (written as 3.14 or 22/7 approximately) is the ratio of the circumference of a circle to its diameter.
The perimeter of a circle is called its circumference: Circumference = 2 x pi x r = pi x d For example, a circle of radius 7 cm has circumference 2 x (22/7) x 7 = 44 cm.
The area of a circle is: Area = pi x r^2 For example, a circle of radius 7 cm has area (22/7) x 49 = 154 square cm.
Composite figures are made by joining two or more simple shapes. To find the area of a composite figure, divide it into recognisable shapes such as rectangles, squares, triangles and semicircles, find the area of each part, and then add (or subtract) them appropriately. For example, the area of a garden that is a rectangle with a semicircular flower bed can be found by adding the area of the rectangle to the area of the semicircle.
Area units are derived from length units by squaring the conversion factor: - 1 m = 100 cm, so 1 square metre = 100 x 100 = 10000 square cm. - 1 km = 1000 m, so 1 square kilometre = 1000 x 1000 = 1000000 square m. - 1 hectare = 10000 square metres.
Volume is measured in cubic units. For a cuboid: Volume = length x breadth x height For a cube of side s: Volume = s^3 - 1 m = 100 cm, so 1 cubic metre = 100 x 100 x 100 = 1000000 cubic cm. - 1 litre = 1000 cubic cm.
| Figure | Perimeter | Area |
|---|---|---|
| Rectangle | 2 x (l + b) | l x b |
| Square | 4 x s | s^2 |
| Triangle | a + b + c | (1/2) x base x height |
| Parallelogram | Sum of four sides | base x height |
| Circle | 2 x pi x r | pi x r^2 |
| Conversion | Relationship |
|---|---|
| Length | 1 m = 100 cm, 1 km = 1000 m |
| Area | 1 square m = 10000 square cm |
| Area | 1 square km = 1000000 square m |
| Area | 1 hectare = 10000 square m |
| Volume | 1 cubic m = 1000000 cubic cm |
| Capacity | 1 litre = 1000 cubic cm |
Perimeter and area give us a complete description of the size of flat figures: how long the boundary is and how much surface is enclosed. The formulas for rectangles, squares, triangles, parallelograms and circles, together with careful unit conversions and the technique of splitting composite figures, equip students to solve real problems of fencing, tiling, painting, sowing and carpeting. The ideas of measurement and units developed here lay the groundwork for surface area and volume of solid shapes in later chapters. Accuracy in applying formulas and converting units is the key to scoring well in mensuration.