Perimeter

How much wire is needed to fence a gar­den? How much lace goes round the edge of a table­cloth? Both ques­tions ask for the length all the way round a shape. That length is called the perime­ter. In this les­son you will learn the perime­ter of rec­tan­gles, squares, tri­an­gles and other poly­gons, how to work back­wards from a perime­ter to a side, and how to find the cost of fenc­ing.

What perime­ter is#

Perime­ter. The perime­ter of a closed fig­ure is the total length of its bound­ary.

For a fig­ure with straight sides, the perime­ter is sim­ply the sum of the lengths of all its sides. Perime­ter is a length, so it is mea­sured in mm, cm, m or km.

Imag­ine an ant walk­ing once round the edge of the shape, start­ing and end­ing at the same cor­ner. The dis­tance it walks is the perime­ter.

Perime­ter of a rec­tan­gle#

A rec­tan­gle has two lengths $l$ and two breadths $b$.

$$\text{perimeter of a rectangle} = l + b + l + b = 2(l + b)$$

Exam­ple 1#

Find the perime­ter of a rec­tan­gle 12 m long and 8 m wide.

$$2(12 + 8) = 2 \times 20 = 40$$

The perime­ter is 40 m.

Perime­ter of a square#

A square has four equal sides. If each side is $a$,

$$\text{perimeter of a square} = 4a$$

Exam­ple 2#

Find the perime­ter of a square of side 7.5 cm.

$$4 \times 7.5 = 30$$

The perime­ter is 30 cm.

Perime­ter of a tri­an­gle and of reg­u­lar poly­gons#

The perime­ter of a tri­an­gle is the sum of its three sides. For an equi­lat­eral tri­an­gle of side $a$ it is $3a$.

A reg­u­lar poly­gon has all its sides equal. If it has $n$ sides, each of length $a$,

$$\text{perimeter} = n \times a$$

A reg­u­lar hexa­gon of side 5 cm has perime­ter $6 \times 5 = 30$ cm.

Exam­ple 3#

The sides of a tri­an­gle are 6.5 cm, 8 cm and 9.5 cm. Find its perime­ter.

$$6.5 + 8 + 9.5 = 24$$

The perime­ter is 24 cm.

Work­ing back­wards#

If you know the perime­ter, you can find a miss­ing side.

Exam­ple 4#

The perime­ter of a rec­tan­gle is 50 m and its length is 15 m. Find its breadth.

$$\begin{aligned} 2(15 + b) &= 50 \\ 15 + b &= 25 \\ b &= 10 \end{aligned}$$

The breadth is 10 m. A quick way: half the perime­ter is one length plus one breadth.

Exam­ple 5#

The perime­ter of a square is 64 cm. Find its side.

$$a = 64 \div 4 = 16$$

The side is 16 cm.

Exam­ple 6: Bend­ing a wire#

A wire is bent into a square of side 9 cm. It is then straight­ened and bent into an equi­lat­eral tri­an­gle. Find the side of the tri­an­gle.

The length of the wire does not change. It equals the perime­ter of the square, $4 \times 9 = 36$ cm. The tri­an­gle has three equal sides, so each side is $36 \div 3 = 12$ cm.

Cost of fenc­ing#

To find the cost of fenc­ing, find the perime­ter and mul­ti­ply by the rate per metre.

$$\text{cost} = \text{perimeter} \times \text{rate}$$

Exam­ple 7#

A field is 120 m long and 80 m wide. Find the cost of fenc­ing it at ₹25 per metre.

$$\text{perimeter} = 2(120 + 80) = 2 \times 200 = 400$$

The perime­ter is 400 m. The cost is $400 \times 25 = 10000$ rupees, that is ₹10,000.

Exam­ple 8: Fenc­ing with a gate#

A square gar­den has side 30 m. A gate 3 m wide is left, and the rest is fenced at ₹40 per metre. Find the cost.

Perime­ter $= 4 \times 30 = 120$ m. Length to be fenced $= 120 - 3 = 117$ m.

Cost $= 117 \times 40 = 4680$ rupees, that is ₹4680.

A shape with a cor­ner cut out#

Exam­ple 9#

A rec­tan­gle 10 cm by 6 cm has a small rec­tan­gle 4 cm by 2 cm cut out of one cor­ner. Find the perime­ter of the shape that is left.

The shape now has six sides. Walk round it. The two full sides that are left are 10 cm and 6 cm. The other two outer sides are now shorter: $10 - 4 = 6$ cm and $6 - 2 = 4$ cm. The cut adds two new edges, of 4 cm and 2 cm.

$$10 + 6 + 6 + 4 + 4 + 2 = 32$$

The perime­ter is 32 cm, which is the same as the perime­ter of the whole rec­tan­gle, $2(10 + 6) = 32$ cm. Cut­ting a rec­tan­gle out of a cor­ner moves two edges inwards but does not change their total length.

Prac­tice#

1. Find the perime­ter of a rec­tan­gle 15 cm long and 9 cm wide.
2. Find the perime­ter of a square of side 12.5 m.
3. Find the perime­ter of a reg­u­lar pen­ta­gon of side 8 cm.
4. The perime­ter of a rec­tan­gle is 72 cm and its breadth is 14 cm. Find its length.
5. The perime­ter of an equi­lat­eral tri­an­gle is 45 cm. Find its side.
6. A wire bent into a rec­tan­gle 14 cm by 10 cm is rebent into a square. Find the side of the square.
7. Find the cost of fenc­ing a rec­tan­gu­lar park 150 m long and 100 m wide at ₹30 per metre.
8. A square plot of side 25 m is to be fenced, leav­ing a gate 4 m wide. Find the cost at ₹50 per metre.
9. A rec­tan­gle 12 cm by 8 cm has a square of side 3 cm cut out of one cor­ner. Find the perime­ter of what is left.

Answers#