Perimeter

The perimeter of a shape is the total distance all the way round its edge. You find it by adding the lengths of all its sides.

Early in a paper, you may be given a shape with its sides marked, or a shape drawn on a centimetre grid, and asked to work out its perimeter, for 1 or 2 marks.

Harder questions, worth 3 to 5 marks, give you the perimeter and ask for a missing side. They may join shapes together, write the sides using a letter such as x, or ask how much fencing to buy and what it costs.

The diagrams are usually not drawn accurately, so don't measure them. Work out every length from the numbers you are given.

Imagine walking all the way round the edge of a shape and back to where you started. The distance you walk is the perimeter.

On a centimetre grid, count the edges of the squares round the outside, not the squares inside. This rectangle has perimeter 5+3+5+3=16 cm. The 15 squares inside are its area, which is a different thing.

Perimeter is a length, so it is measured in mm, cm or m. Area is measured in square units, such as \text{cm}^2

Start at one corner and go round the shape in order, ticking each side as you add it. Then no side is missed and none is counted twice.

The opposite sides of a rectangle are equal, so a diagram often marks only the length and the width. The perimeter is two lengths and two widths:

P=2l+2w

where l is the length and w is the width.

All four sides of a square are equal, so its perimeter is 4 times one side. With sides of 8 m, the perimeter is 4\times8=32 m.

In a regular shape, all the sides are the same length. A regular pentagon has 5 equal sides and a regular hexagon has 6. Multiply one side by the number of sides.

Tick marks on sides show that they are equal. So a side with the same tick mark as a 6 cm side is also 6 cm long.

Use one unit for every side. To change mm to cm, divide by 10. For example, a regular hexagon with sides of 45 mm has perimeter 6\times45=270 mm, which is 27 cm.

In an L-shape every corner is a right angle, so each side goes straight across or straight up and down. It has six sides, and usually some of them are not marked.

Find each missing side from the sides you know. The sides across the top add up to the side across the bottom, and the heights on one side add up to the height on the other. If the whole width is 10 m and one part of the top is 6 m, the other part is 10-6=4 m.

When shapes are joined, the edges where they touch are inside the new shape, so they are not part of its perimeter. Don't just add the perimeters of the separate pieces.

Each square has perimeter 20 cm, but the joined shape has perimeter 10+5+10+5=30 cm, not 40 cm. The two 5 cm edges that touch are inside.

Sometimes you are given the perimeter and asked for a missing side. Add the sides you know, then subtract from the perimeter. For example, a triangle with perimeter 30 cm and sides of 8 cm and 13 cm has a third side of 30-8-13=9 cm.

For a rectangle, take off both widths first, then halve what is left to get one length. Only divide the perimeter by 4 when the shape is a square.

When the sides are written with a letter, add them to make an expression for the perimeter. Collect like terms, make it equal to the perimeter you are given, then solve the equation.

Finding x is often not the end. Read the question again: it may want the length of one side, so put your value of x back in.

Fencing, edging, rope and ribbon go round the edge of something, so these are perimeter questions. Turf, paint and tiles cover a surface, so those are area questions.

Read which sides need fencing. Leave out a side along a wall, and subtract the width of a gate.

Fencing is often sold in panels or rolls of one length. Divide the length you need by the length of one panel, then round up, because you can't buy part of one. Then work out the cost.

• Adding only the numbers you can see. Work out every side that is not marked, then add all of them.
• Adding the perimeters of the separate pieces. Edges where shapes touch are inside the shape, so leave them out.
• Working out the area instead. Perimeter is the distance round the edge, so add the sides; don't multiply them.
• Dividing a rectangle's perimeter by 4. Take off the two sides you know, then halve what is left.
• Stopping before the question is answered. If you have found x or a perimeter, check whether the question wants a side, a number of panels or a cost.
• Making one side equal to the perimeter. Add all the sides to make the perimeter first, then make that equal to the total.