Perimeter, area and volume

Perimeter

Perimeter is the total distance around the outside of a 2-D shape, measured in units of length (cm, m). For rectilinear shapes, made only of right angles, some side lengths are often missing and have to be worked out from the others.

• Missing horizontal side = 12 − 7 = 5 cmOpposite sides of a rectilinear shape balance out.
• Perimeter: add all six sides, ticking each as you goMarking sides stops one being missed.

The classic error is adding only the labelled sides. Asking children to label every side, including the missing ones, before adding fixes this.

Area

Area is the amount of surface a shape covers, measured in square units (cm², m²). Year 6 pupils use three formulae:

The formulae make sense once children see where they come from. Cut a triangle off one end of a parallelogram and move it to the other end, and it becomes a rectangle with the same base and height. Two identical triangles make a parallelogram, which is why a triangle's area is half.

• Triangle, base 10 cm, height 6 cm: ½ × 10 × 6 = 30 cm²Use the perpendicular height.
• Triangle: 10 × 6 = 60 cm²Forgetting to halve.
• Parallelogram: base × slanted sideThe height must be perpendicular to the base.

For compound shapes, split into rectangles (or triangles), find each area and add. Year 6 pupils also need to know that shapes with the same area can have different perimeters, and the other way round.

Volume

Volume is the amount of space a 3-D object takes up, measured in cubic units (cm³, m³). The volume of a cuboid is length × width × height. Capacity is how much a container holds, usually in millilitres or litres; 1 cm³ = 1 ml.

• Cuboid 5 cm × 4 cm × 3 cm = 60 cm³Multiply all three dimensions.
• Cuboid 5 + 4 + 3 = 12 cm³Adding instead of multiplying.

Building cuboids from centimetre cubes and counting them shows why the formula works: one layer is length × width, and there are “height” layers.

In the SATs

These topics appear on the reasoning papers. Typical questions include finding the perimeter of a rectilinear shape with missing lengths, finding the area of a triangle or parallelogram drawn on a grid or labelled with measurements, comparing shapes with equal area or perimeter, and finding or comparing volumes of cuboids. Units matter: an answer in cm when the question wants cm² can lose the mark.

Common mistakes

• Confusing area and perimeter.
• Missing sides when adding a perimeter.
• Using the slanted side instead of the perpendicular height.
• Forgetting to halve for triangles.
• Adding dimensions for volume, or using the wrong units (cm instead of cm² or cm³).

Teaching ideas

Fence and lawn

Perimeter is the fence; area is the grass. A simple image children return to when they're unsure which measure a question wants.

Cut and rearrange

Children cut out a paper parallelogram, snip off a triangle and move it to make a rectangle. Then they cut a parallelogram along its diagonal to make two triangles. The formulae stop being rules to memorise.

Same area, different perimeter

Give pupils 24 squares and ask them to make as many different rectangles as they can. Record the perimeter of each. The results surprise most children.

Build the volume

Give a cuboid's dimensions and ask pupils to build it with cubes, then check the formula. Then work backwards: build a cuboid with a volume of 36 cm³.

Try it: five quick questions

SATs-style questions to use as a starter or a quick check. Tap to see each answer.

1. A rectangle is 9 cm by 4 cm. Find its perimeter and area.

   Show answer

   Perimeter 26 cm, area 36 cm².

2. Find the area of a triangle with base 8 cm and perpendicular height 5 cm.

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   20 cm²

3. Find the area of a parallelogram with base 7 m and perpendicular height 3 m.

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   21 m²

4. Find the volume of a cuboid 6 cm × 5 cm × 2 cm.

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   60 cm³

5. A cuboid has a volume of 48 cm³. Its base is 4 cm by 3 cm. How tall is it?

   Show answer

   4 cm