Perimeter, perimeter formulas

Build a shape and find its perimeter

Click the squares to build a shape, or drag across them. Click a square again to take it away. Every small square is 1 inch long on each side. Press Count and watch the orange dot walk round the shape and count its sides. Press Play for tasks.

What Is Perimeter?

Walk all the way round a park and you walk its perimeter. On this page you will learn what perimeter is, how to count it on squared paper and how to find the perimeter of a triangle, a square, a rectangle and any other shape with straight sides.

On this page: Perimeter game · Perimeter · Squared paper · Triangle · Square · Rectangle · More sides · Missing side · Units · Perimeter and area · Word problems · Practice · Older students

Perimeter

An ant walks along the edge of a leaf until it comes back to where it started. A fence goes all the way round a garden. A ribbon goes round the edge of a picture. Each time we go once all the way round a shape.

We write the perimeter with the letter P. The perimeter is a length, so we measure it in inches, feet, centimeters, meters and other units of length.

Counting the perimeter on squared paper

Here every small square is 1 inch long on each side. Start at the orange dot and walk round the shape. Count every small side you pass.

We counted 12 small sides, so the perimeter of the rectangle is 12 inches. Count the small sides along the edge, not the squares inside the shape.

This shape has 14 small sides on its edge, so its perimeter is 14 inches.

Tip: put your finger on a corner, count while you go round, and stop when you are back at the same corner. In the game at the top of this page the button Count does this for you.

Measuring the sides with a ruler

When a shape is not on squared paper, measure each side with a ruler. Then add the lengths.

P = 3 in + 4 in + 5 in = 12 in

The perimeter of this triangle is 12 inches.

Perimeter of a triangle

A triangle has 3 sides. Call their lengths a, b and c.

Example. A triangle has sides of 6 cm, 5 cm and 4 cm. What is its perimeter?

P = 6 cm + 5 cm + 4 cm = 15 cm

Equilateral triangle

In an equilateral triangle all three sides are equal. The small marks on the sides show that they are equal.

Example. Each side of an equilateral triangle is 7 cm. Then P = 3 × 7 cm = 21 cm.

Perimeter of a square

A square has 4 equal sides. Let each side be a long.

Example. A square has sides of 5 cm.

P = 5 cm + 5 cm + 5 cm + 5 cm = 20 cm

It is quicker to multiply: P = 4 × 5 cm = 20 cm.

Perimeter of a rectangle

A rectangle has two long sides and two short sides. The long side is the length, and the short side is the width. The opposite sides are equal. Let the length be a and the width be b.

Example. A rectangle is 6 cm long and 4 cm wide. Find its perimeter.

Add the sides: P = 6 cm + 4 cm + 6 cm + 4 cm = 20 cm.

Or double each side: P = 2 × 6 cm + 2 × 4 cm = 12 cm + 8 cm = 20 cm.

Or add the length and the width, then double: P = 2 × (6 cm + 4 cm) = 2 × 10 cm = 20 cm.

All three ways give the same answer. Use the one you like best.

Shapes with more sides

The rule is the same for every shape with straight sides: add all the sides. Count the sides first, so that you do not miss one.

This house shape has 5 sides:

P = 6 cm + 5 cm + 4 cm + 4 cm + 5 cm = 24 cm

Finding a missing side

Sometimes we know the perimeter and have to find a side.

Example 1. A triangle has a perimeter of 20 cm. Two of its sides are 6 cm and 8 cm. How long is the third side?

The two sides together are 6 cm + 8 cm = 14 cm. The third side is the rest of the perimeter: 20 cm − 14 cm = 6 cm.

Example 2. The perimeter of a square is 24 cm. How long is each side?

The four sides are equal, so divide by 4: 24 cm ÷ 4 = 6 cm.

Example 3. The perimeter of a rectangle is 18 cm. Its length is 5 cm. What is its width?

The length and the width together make half of the perimeter: 18 cm ÷ 2 = 9 cm. So the width is 9 cm − 5 cm = 4 cm.

Check: 5 cm + 4 cm + 5 cm + 4 cm = 18 cm. Correct!

Sides you can work out

The floor of a room is made of two rectangles. Two of its sides have no numbers. We can work them out from the other sides.

• The left side is 6 ft and the right side is 2 ft. So the side in the middle that goes down is 6 ft − 2 ft = 4 ft.
• The bottom side is 8 ft and the top side is 3 ft. So the side in the middle that goes across is 8 ft − 3 ft = 5 ft.

Now add all six sides: P = 3 ft + 4 ft + 5 ft + 2 ft + 8 ft + 6 ft = 28 ft.

Use the same units

Before you add the sides, make sure they are all in the same unit. Remember: 1 foot = 12 inches and 1 meter = 100 centimeters.

Example. A rectangle is 2 feet long and 8 inches wide. Find its perimeter in inches.

First change feet to inches: 2 ft = 24 in. Then P = 24 in + 8 in + 24 in + 8 in = 64 in.

Perimeter is not area

The perimeter is the length of the edge: we count the sides round the shape. The area is how much space is inside the shape: we count the squares that cover it. Area is measured in square units, for example square inches (sq in).

Both shapes are made of 4 squares, so they have the same area: 4 square inches. But their perimeters are different. Shape A has a perimeter of 10 inches, and shape B has a perimeter of 8 inches.

Try it in the game at the top of this page: build different shapes from the same number of squares and compare their perimeters.

Perimeter in real life

We need the perimeter whenever something goes all the way round: a fence round a garden, a frame round a picture, a border round a bulletin board, lace round a tablecloth.

Problem. A garden is 9 ft long and 5 ft wide. How many feet of fence do we need to go all the way round it?

The fence goes round the garden, so we need the perimeter:

P = 9 ft + 5 ft + 9 ft + 5 ft = 28 ft

Answer: we need 28 feet of fence.

Practice

Press Play in the game at the top of this page, or try these practice tests:

Questions

1. A triangle has sides of 3 cm, 5 cm and 6 cm. What is its perimeter?

P = 3 cm + 5 cm + 6 cm = 14 cm.

2. A square has sides of 9 inches. What is its perimeter?

P = 4 × 9 in = 36 in.

3. A rectangle is 8 m long and 3 m wide. What is its perimeter?

P = 8 m + 3 m + 8 m + 3 m = 22 m.

4. Each side of an equilateral triangle is 10 cm. What is its perimeter?

P = 3 × 10 cm = 30 cm.

5. The perimeter of a square is 32 cm. How long is one side?

32 cm ÷ 4 = 8 cm.

6. A triangle has a perimeter of 25 in. Two of its sides are 7 in and 10 in. How long is the third side?

7 in + 10 in = 17 in, and 25 in − 17 in = 8 in.

7. The perimeter of a rectangle is 30 cm and its length is 10 cm. What is its width?

Half of the perimeter is 30 cm ÷ 2 = 15 cm, so the width is 15 cm − 10 cm = 5 cm.

8. A picture is 20 cm long and 15 cm wide. How much ribbon do we need to go once round its edge?

P = 20 cm + 15 cm + 20 cm + 15 cm = 70 cm. We need 70 cm of ribbon.

9. A rectangle is 1 foot long and 4 inches wide. What is its perimeter in inches?

1 ft = 12 in, so P = 12 in + 4 in + 12 in + 4 in = 32 in.

10. Mia makes a rectangle from 6 squares. Leo makes a different rectangle from the same 6 squares. Do the two rectangles have the same perimeter?

No. A row of 6 squares is a 6 by 1 rectangle with a perimeter of 14 units. Two rows of 3 squares make a 3 by 2 rectangle with a perimeter of 10 units. The area is the same, 6 squares, but the perimeters are different.

Perimeter formulas for older students

These shapes are studied in higher grades. The rule is still the same: add all the sides.

Perimeter of a parallelogram

The opposite sides of a parallelogram are equal, as in a rectangle:

P = a + b + a + b = 2 × a + 2 × b

Perimeter of a rhombus

A rhombus has four equal sides, as a square does:

P = 4 × a

Perimeter of an isosceles trapezoid

Let a and b be the parallel sides (the bases). In an isosceles trapezoid the other two sides (the legs) are equal; let each be c.

P = a + b + c + c = a + b + 2 × c

Circumference of a circle

The perimeter of a circle is called its circumference C. With the radius r and the diameter d = 2r:

$$C = \pi d = 2\pi r$$

Here π ≈ 3.14.

Regular polygon

A regular polygon with n sides has all its sides equal. If each side is a, then P = n × a. If we know the distance R from the centre to a vertex instead:

$$P = 2nR\sin\frac{180^\circ}{n}$$