Geometry · Grades 3, 4

Perimeter: The Distance Around a Shape

Quick answer

Perimeter is the total distance around the outside of a shape, found by adding the lengths of all its sides. A rectangle 8 by 5 has a perimeter of 8 + 5 + 8 + 5 = 26. The shortcut is 2 times length plus 2 times width, because opposite sides of a rectangle are equal. Perimeter is a length, so it is measured in centimetres or metres, never square units.

What you'll learn

  • Find the perimeter of any shape by adding its sides
  • Use the rectangle shortcut and explain where it comes from
  • Find a missing side length from a known perimeter

Walk around the edge

Perimeter is the distance around the outside of a shape.

Imagine walking right around the edge and back to where you started. How far did you walk? That is the perimeter, and it is found by adding the lengths of all the sides.

A rectangle 8 cm by 5 cm A rectangle labelled 8 cm along the bottom edge and 5 cm up the left edge. 8 cm 5 cm
A rectangle 8 cm by 5 cm
8+5+8+5=26 cm8 + 5 + 8 + 5 = 26 \text{ cm}

Four sides, all added. There is nothing more to the definition, and it works on any shape with straight sides — a triangle, a hexagon, or an odd L-shaped room.

Why the shortcut works

A rectangle has two pairs of equal sides. So you can double two numbers instead of adding four.

P=2l+2wP = 2l + 2w

For the rectangle above: 2(8)+2(5)=16+10=262(8) + 2(5) = 16 + 10 = 26 cm. The same answer.

It has to be. The shortcut is the same addition, grouped differently:

8+8⏟2l+5+5⏟2w=2l+2w\underbrace{8 + 8}_{2l} + \underbrace{5 + 5}_{2w} = 2l + 2w

A square has four equal sides. So its perimeter is P=4sP = 4s, for the same reason.

Perimeter and area are different questions

Mixing these two up costs more marks than any slip in the adding. Sort them out now.

MeasuresUnitsThink of
perimeterdistance aroundcm, mthe fence
areaspace insidecm², m²the field

Every shape has both. They answer different questions.

Two shapes can share a perimeter and still hold very different amounts of space. A 1×111 \times 11 rectangle and a 6×66 \times 6 square both have a perimeter of 2424. But their areas are 1111 and 3636.

Working backwards

If you know the perimeter but not a side, take away what you do know.

A rectangle has a perimeter of 3030 cm and a length of 99 cm.

The two lengths use up 1818 cm. That leaves 1212 cm for the two widths. So each width is 66 cm.

That is the same undoing you do when solving an equation. It is one:

2(9)+2w=30⇒2w=12⇒w=62(9) + 2w = 30 \quad\Rightarrow\quad 2w = 12 \quad\Rightarrow\quad w = 6

Worked examples

Common mistakes

Practice problems

  1. Find the perimeter of a rectangle 1010 cm by 44 cm.

    Answer

    2828 cm

    Full solution

    2(10)+2(4)=20+8=282(10) + 2(4) = 20 + 8 = 28 cm.

  2. Find the perimeter of a square with sides of 77 m.

    Answer

    2828 m

    Full solution

    4(7)=284(7) = 28 m.

  3. A triangle has sides 55, 99 and 1212 cm. Find its perimeter.

    Answer

    2626 cm

    Full solution

    5+9+12=265 + 9 + 12 = 26 cm.

  4. Find the perimeter of a rectangle 2.52.5 m by 1.51.5 m.

    Answer

    88 m

    Full solution

    2(2.5)+2(1.5)=5+3=82(2.5) + 2(1.5) = 5 + 3 = 8 m.

  5. A rectangle has a perimeter of 3434 cm and a length of 1111 cm. Find its width.

    Hint

    Take the two lengths off the total first.

    Answer

    66 cm

    Full solution

    The two lengths account for 2222 cm, leaving 1212 cm for the two widths, so each is 66 cm.

    Checking: 2(11)+2(6)=342(11) + 2(6) = 34 ✓

  6. A square has a perimeter of 4848 cm. How long is each side?

    Answer

    1212 cm

    Full solution

    All four sides are equal, so 48÷4=1248 \div 4 = 12 cm.

  7. A hexagon has six equal sides of 55 cm. Find its perimeter.

    Answer

    3030 cm

    Full solution

    6×5=306 \times 5 = 30 cm. Equal sides mean multiplication does the adding for you.

  8. A field 4040 m by 2525 m is fenced, but a 66 m gateway is left open. How much fencing is needed?

    Answer

    124124 m

    Full solution

    The perimeter is 2(40)+2(25)=1302(40) + 2(25) = 130 m, and the gateway removes 66 m: 130−6=124130 - 6 = 124 m.

  9. A 1×81 \times 8 rectangle and a 3×63 \times 6 rectangle — which has the bigger perimeter, and which the bigger area?

    Hint

    Work out both quantities for both shapes.

    Answer

    The 1×81 \times 8 has the bigger perimeter; the 3×63 \times 6 has the bigger area.

    Full solution

    Perimeters: 2(1)+2(8)=182(1) + 2(8) = 18 against 2(3)+2(6)=182(3) + 2(6) = 18 — actually equal.

    Areas: 1×8=81 \times 8 = 8 against 3×6=183 \times 6 = 18.

    So the perimeters tie and the areas differ by more than double. Perimeter tells you almost nothing about area, which is the point worth taking from this.

  10. Mia says a rectangle 66 cm by 44 cm has a perimeter of 2424 cm² . Find both errors.

    Hint

    Check the number and the units separately.

    Answer

    She found the area, and gave it the wrong name. The perimeter is 2020 cm.

    Full solution

    6×4=246 \times 4 = 24 is the area, in cm². The perimeter is 2(6)+2(4)=202(6) + 2(4) = 20 cm.

    The units gave her away: square units can only belong to area, so “cm²” and “perimeter” cannot appear in the same answer.

Frequently asked questions

What is perimeter?

The total distance around the outside of a shape. Walk right around the edge and back to where you started, and the distance you covered is the perimeter.

How do I find the perimeter of a rectangle?

Add all four sides, or use the shortcut 2 times the length plus 2 times the width. Both give the same answer, because a rectangle's opposite sides are equal.

What units does perimeter use?

Ordinary length units — centimetres, metres, inches, feet. It is a distance, so it never takes square units. Those belong to area.

Can two shapes have the same perimeter but different areas?

Yes, and it happens constantly. A 1 by 11 rectangle and a 6 by 6 square both have a perimeter of 24, but their areas are 11 and 36. Perimeter is the fence; area is the field.

How do I find a missing side if I know the perimeter?

Add up the sides you know and subtract that from the perimeter. What is left is the missing side, or the total of the missing sides if there is more than one.

What to learn next

Formulas on this page

Key terms in this lesson

Area
Area is the amount of space inside a flat shape, counted in square units such as cm². It comes from multiplying two lengths, which is why the units are squared.

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

  • CCSS.MATH.CONTENT.3.MD.D.8Measurement and DataSolve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.
  • CCSS.MATH.CONTENT.4.MD.A.3Measurement and DataApply the area and perimeter formulas for rectangles in real world and mathematical problems.
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