Year 6: Same area, different perimeters, and when to use a formula
Part of the Year 6 area, volume and measures chapter. Chapter overview · whole chapter on one page
Same area, different perimeters, and when to use a formula
Two shapes can have the same area and different perimeters. Look at the rectangles with an area of 36 cm²: a long thin rectangle has a big perimeter, and the square has the smallest perimeter.
| Length (cm) | Width (cm) | Area (cm²) | Perimeter (cm) |
|---|---|---|---|
| 1 | 36 | 36 | 74 |
| 2 | 18 | 36 | 40 |
| 3 | 12 | 36 | 30 |
| 4 | 9 | 36 | 26 |
| 6 | 6 | 36 | 24 |
The reverse is also true: shapes with the same perimeter can have different areas. Cutting a corner out of a rectangle keeps the perimeter the same, but makes the area smaller.
A formula can be used for the standard shapes: a rectangle, a square, a parallelogram, a triangle and a cuboid. For any other shape, such as an L-shape, first split it into standard shapes, then use the formula for each part.
Key ideas
- Shapes with the same area can have different perimeters.
- Shapes with the same perimeter can have different areas.
- Cutting a corner out of a rectangle keeps the perimeter the same and makes the area smaller.
- Use a formula for a standard shape (rectangle, square, parallelogram, triangle, cuboid). For any other shape, split it into standard shapes first.
Key words. The perimeter is the total distance all the way round the outside of a shape. Add the lengths of all the outside sides.
Method. List the rectangles with this area, work out each perimeter, and choose the smallest.
- Find every pair of whole-number sides that multiply to make the area. The sides must multiply to 24.1 × 24, 2 × 12, 3 × 8, 4 × 6
- Find the perimeter of each rectangle. Perimeter = 2 times (length + width).1 × 24: 2 × (1 + 24) = 50 ; 2 × 12: 2 × (2 + 12) = 28 ; 3 × 8: 2 × (3 + 8) = 22 ; 4 × 6: 2 × (4 + 6) = 20
- Choose the smallest perimeter. The smallest number in the list is the answer.The smallest is 20
- Write the answer. The perimeter is in centimetres.20
Check. Every pair was checked, so no rectangle is missed. ✓
Watch out. A square has the smallest perimeter for its area. Rectangles that are long and thin have big perimeters.
Key words. The area is the space inside a flat shape, in square units (cm²). The perpendicular height is measured straight up from the base, at a right angle to it. It is not the slanted side.
Method. Find the area of the whole rectangle, then take away the piece that is cut out.
- Find the area of the whole rectangle. Area is length times width.8 × 5 = 40
- Find the area of the corner that is cut out, 3 cm by 2 cm. The cut-out piece is a rectangle too.3 × 2 = 6
- Take the cut-out area away. The shape that is left has the rest of the area.40 - 6 = 34
- Write the answer. The area is in square centimetres.34
Check. Add the two parts back: 6 + 34 = 40. ✓
Watch out. Do not add the two areas. The cut-out piece is taken away.
Key words. The perimeter is the total distance all the way round the outside of a shape. Add the lengths of all the outside sides.
Method. Work out the area of every rectangle with this perimeter, and choose the largest.
- Find each pair of whole-number sides that makes this perimeter. Each pair must add up to 12.1 + 11 = 12, 2 + 10 = 12, 3 + 9 = 12, 4 + 8 = 12, 5 + 7 = 12, 6 + 6 = 12
- Find the area of each rectangle. Area is length times width.1 × 11 = 11 ; 2 × 10 = 20 ; 3 × 9 = 27 ; 4 × 8 = 32 ; 5 × 7 = 35 ; 6 × 6 = 36
- Choose the biggest area. The biggest number in the list is the answer.The biggest area is 36
- Write the answer. The area is in square centimetres.36
Check. The rectangle with sides 6 cm and 6 cm has area 6 × 6 = 36 cm². ✓
Watch out. Shapes with the same perimeter can have different areas. The most square shape has the biggest area.
Watch out
Common mistake: thinking that the same area means the same perimeter. A 1 cm by 36 cm strip has an area of 36 cm² but a perimeter of 74 cm. Always work out the perimeter from the sides.
Exam tip
When a shape is not a rectangle, split it into rectangles. Work out the area of each part and add them together.
Exercise 9E Area, perimeter and formulae
Non-calculatorKey words. The perimeter is the total distance all the way round the outside of a shape. Add the lengths of all the outside sides.
Method. Add the length and the width, then multiply by 2.
- Write the formula. Perimeter = 2 times (length + width).P = 2 × (l + w)
- Add the length and the width. This is half of the distance round the shape.9 + 4 = 13
- Multiply by 2. The shape has two lengths and two widths.2 × 13 = 26
- Write the answer. The perimeter is in centimetres.26
Check. 9 + 4 + 9 + 4 = 26. ✓
Watch out. Do not forget to count all four sides.
Key words. The area is the space inside a flat shape, in square units (cm²). The perpendicular height is measured straight up from the base, at a right angle to it. It is not the slanted side.
Method. Multiply the length by the width.
- Write the formula. Area is length times width.A = l × w
- Multiply the length by the width. The area is in square centimetres.9 × 4 = 36
- Write the answer. 36
Check. Turn the rectangle round: 4 × 9 = 36, the same answer. ✓
Watch out. Area is measured in square units, so it is a multiplication, not an addition.
Key words. The area is the space inside a flat shape, in square units (cm²). The perpendicular height is measured straight up from the base, at a right angle to it. It is not the slanted side.
Method. Find every pair of whole-number sides that makes this area, and count them without counting a shape twice.
- List the pairs of whole-number sides that multiply to make the area. A 1 by 36 rectangle is the same shape as a 36 by 1 rectangle.1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6
- Count the pairs. Each pair is one different rectangle.There are 5 pairs
- Write the answer. 5
Check. Each pair multiplies to 36: check the first and the last pair. ✓
Watch out. Do not count 36 by 1 and 1 by 36 as two different rectangles.
Key words. The perimeter is the total distance all the way round the outside of a shape. Add the lengths of all the outside sides.
Method. Add the length and the width, then multiply by 2.
- Write the formula. Perimeter = 2 times (length + width).P = 2 × (l + w)
- Add the length and the width. This is half of the distance round the shape.6 + 6 = 12
- Multiply by 2. The shape has two lengths and two widths.2 × 12 = 24
- Write the answer. The perimeter is in centimetres.24
Check. 6 + 6 + 6 + 6 = 24. ✓
Watch out. Do not forget to count all four sides.
Key words. The perimeter is the total distance all the way round the outside of a shape. Add the lengths of all the outside sides.
Method. Work out the area of every rectangle with this perimeter, and choose the largest.
- Find each pair of whole-number sides that makes this perimeter. Each pair must add up to 10.1 + 9 = 10, 2 + 8 = 10, 3 + 7 = 10, 4 + 6 = 10, 5 + 5 = 10
- Find the area of each rectangle. Area is length times width.1 × 9 = 9 ; 2 × 8 = 16 ; 3 × 7 = 21 ; 4 × 6 = 24 ; 5 × 5 = 25
- Choose the biggest area. The biggest number in the list is the answer.The biggest area is 25
- Write the answer. The area is in square centimetres.25
Check. The rectangle with sides 5 cm and 5 cm has area 5 × 5 = 25 cm². ✓
Watch out. Shapes with the same perimeter can have different areas. The most square shape has the biggest area.
Key words. The area is the space inside a flat shape, in square units (cm²). The perpendicular height is measured straight up from the base, at a right angle to it. It is not the slanted side.
Method. Find the area of the whole rectangle, then take away the piece that is cut out.
- Find the area of the whole rectangle. Area is length times width.10 × 6 = 60
- Find the area of the corner that is cut out, 4 cm by 2 cm. The cut-out piece is a rectangle too.4 × 2 = 8
- Take the cut-out area away. The shape that is left has the rest of the area.60 - 8 = 52
- Write the answer. The area is in square centimetres.52
Check. Add the two parts back: 8 + 52 = 60. ✓
Watch out. Do not add the two areas. The cut-out piece is taken away.
Key words. The perimeter is the total distance all the way round the outside of a shape. Add the lengths of all the outside sides.
Method. Add up the lengths of all the outside sides.
- Name every outside side. Go all the way round the outside. Each side of the cut-out goes on the outside too.6, 2, 4, 4, 10, 6 cm
- Add the six outside sides. This gives the total distance round the shape.6 + 2 + 4 + 4 + 10 + 6 = 32
- Write the answer. The perimeter is in centimetres.32
Check. The same as the whole rectangle: 2 × (10 + 6) = 32 cm. ✓
Watch out. The cut moves some sides but keeps the total. Count the short sides of the cut as well.
Key words. The perimeter is the total distance all the way round the outside of a shape. Add the lengths of all the outside sides.
Method. Work out the perimeter and the area of each rectangle, then compare the perimeters.
- Find the perimeter of rectangle A. Add the length and the width, then multiply by 2.2 × (5 + 6) = 22 cm
- Find the perimeter of rectangle B. Both rectangles have an area of 30 cm², but B is much longer.2 × (3 + 10) = 26 cm
- Find the difference between the two perimeters. Take the smaller perimeter away from the bigger one.26 - 22 = 4
- Write the answer. The difference is in centimetres.4
Check. 22 + 4 = 26. Both areas are 30 cm², because 5 × 6 = 30 and 3 × 10 = 30. ✓
Watch out. Equal areas do not mean equal perimeters. Work out each perimeter separately.
Key words. The area is the space inside a flat shape, in square units (cm²). The perpendicular height is measured straight up from the base, at a right angle to it. It is not the slanted side.
Method. Find the area of each shape, then find the difference.
- Find the area of the 5 cm by 7 cm rectangle. Area is length times width.5 × 7 = 35 cm²
- Find the area of the 6 cm by 6 cm square. Both shapes have the same perimeter of 24 cm, but the square has a bigger area.6 × 6 = 36 cm²
- Find the difference between the two areas. Take the smaller area away from the bigger one.36 - 35 = 1
- Write the answer. The difference is in square centimetres.1
Check. 5 + 7 + 5 + 7 = 24 and 6 + 6 + 6 + 6 = 24, so both perimeters are 24 cm. ✓
Watch out. Same perimeter does not mean same area. The most square shape has the biggest area.
Key words. The area is the space inside a flat shape, in square units (cm²). The perpendicular height is measured straight up from the base, at a right angle to it. It is not the slanted side.
Method. Find the area of the whole rectangle, then take away the piece that is cut out.
- Find the area of the whole rectangle. Area is length times width.7 × 5 = 35
- Find the area of the corner that is cut out, 2 cm by 2 cm. The cut-out piece is a rectangle too.2 × 2 = 4
- Take the cut-out area away. The shape that is left has the rest of the area.35 - 4 = 31
- Write the answer. The area is in square centimetres.31
Check. Add the two parts back: 4 + 31 = 35. ✓
Watch out. Do not add the two areas. The cut-out piece is taken away.
Key words. The perimeter is the total distance all the way round the outside of a shape. Add the lengths of all the outside sides.
Method. Work out the perimeter of the whole rectangle, then of the L-shape, and compare them.
- Find the perimeter of the whole rectangle. It is 7 cm by 5 cm before the corner is cut out.2 × (7 + 5) = 24 cm
- Add the six outside sides of the L-shape. The sides of the cut-out go on the outside, so they are counted.5 + 2 + 2 + 3 + 7 + 5 = 24 cm
- Compare the two perimeters. The two perimeters are the same.24 = 24
- Write the answer. Ali is not right, because the perimeter does not get smaller.No
Check. Cutting the corner moves two sides but keeps the total length the same. ✓
Watch out. The perimeter stays the same when a corner is cut out. Only the area gets smaller.
Key words. The perimeter is the total distance all the way round the outside of a shape. Add the lengths of all the outside sides.
Method. List the perimeters of every rectangle with whole-number sides and an area of 36 cm².
- List the rectangles with an area of 36 cm². The sides must multiply to 36.1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6
- Find the perimeter of each one. Perimeter = 2 times (length + width).74, 40, 30, 26, 24
- Check whether 25 cm is in the list. If it is not there, no such rectangle exists with whole-number sides.25 is not in the list
- Write the answer. A rectangle with whole-number sides cannot have these measurements.No
Check. 25 is odd, but 2 × (length + width) is always even, so no rectangle can have a perimeter of 25 cm. ✓
Watch out. The perimeter of a rectangle with whole-number sides is always an even number.