Year 6 Maths reasoning (SATs): Same area, different perimeters, and volume of cuboids
Part of the Year 6 Maths reasoning (SATs) measures in context chapter. Chapter overview · whole chapter on one page
Same area, different perimeters, and volume of cuboids
Rule
Rectangles with the same area can have different perimeters. The closer the length and width, the smaller the perimeter.
Volume of a cuboid = length x width x height. Cubic units are written as cm3.
Key ideas
- Same area, different perimeter: a square has the smallest perimeter for a given area.
- Perimeter = 2 x (length + width).
- Volume is the number of unit cubes that fill the box.
Method. List the pairs that multiply to the area, then find each perimeter.
- List the whole-number pairs that multiply to the area. Each pair of sides gives area 24.1 x 24; 2 x 12; 3 x 8; 4 x 6
- Work out each perimeter. Perimeter = 2 x (length + width).1 by 24 gives 50; 2 by 12 gives 28; 3 by 8 gives 22; 4 by 6 gives 20
- Choose the smallest perimeter. The smallest perimeter is the one with the sides closest in length.4 by 6 (perimeter 20)
- Write the answer. 4 by 6 (perimeter 20)
Check. The closest pair of sides gives the smallest perimeter. ✓
Watch out. The same area can have very different perimeters: a long thin rectangle has a bigger perimeter than a square.
Method. Multiply the length, width and height.
- Write the formula. Volume of a cuboid = length x width x height.4 x 3 x 5
- Multiply. Multiply the three lengths.4 x 3 = 12, then 12 x 5 = 60
- Write the units. Cubic units for volume.60 cm³
- Write the answer. 60 cm³
Check. 4 x 3 x 5 = 60. ✓
Watch out. Volume is measured in cubic units, so the answer ends with a small 3, such as cm³.
Method. Work out the volume of each shape.
- Cube. Volume = 4 x 4 x 4 = 64 cm3.64 cm³
- Cuboid. Volume = 6 x 4 x 3 = 72 cm3.72 cm³
- Compare. 72 is more than 64, so the cuboid has the bigger volume.cuboid
- Write the answer. cuboid
Check. 72 > 64. ✓
Watch out
Watch out. A long thin rectangle can have a much bigger perimeter than a square with the same area.
Exam tip
Practice guide. List the pairs of sides that multiply to the area, then work out each perimeter.
Exercise 4E Practice: same area and volume
Non-calculatorMethod. List the pairs that multiply to the area, then find each perimeter.
- List the whole-number pairs that multiply to the area. Each pair of sides gives area 36.1 x 36; 2 x 18; 3 x 12; 4 x 9; 6 x 6
- Work out each perimeter. Perimeter = 2 x (length + width).1 by 36 gives 74; 2 by 18 gives 40; 3 by 12 gives 30; 4 by 9 gives 26; 6 by 6 gives 24
- Choose the smallest perimeter. The smallest perimeter is the one with the sides closest in length.6 by 6 (perimeter 24)
- Write the answer. 6 by 6 (perimeter 24)
Check. The closest pair of sides gives the smallest perimeter. ✓
Watch out. The same area can have very different perimeters: a long thin rectangle has a bigger perimeter than a square.
Method. List the pairs that multiply to the area, then find each perimeter.
- List the whole-number pairs that multiply to the area. Each pair of sides gives area 20.1 x 20; 2 x 10; 4 x 5
- Work out each perimeter. Perimeter = 2 x (length + width).1 by 20 gives 42; 2 by 10 gives 24; 4 by 5 gives 18
- Choose the smallest perimeter. The smallest perimeter is the one with the sides closest in length.4 by 5 (perimeter 18)
- Write the answer. 4 by 5 (perimeter 18)
Check. The closest pair of sides gives the smallest perimeter. ✓
Watch out. The same area can have very different perimeters: a long thin rectangle has a bigger perimeter than a square.
Method. Multiply the length, width and height.
- Write the formula. Volume of a cuboid = length x width x height.6 x 2 x 5
- Multiply. Multiply the three lengths.6 x 2 = 12, then 12 x 5 = 60
- Write the units. Cubic units for volume.60 cm³
- Write the answer. 60 cm³
Check. 6 x 2 x 5 = 60. ✓
Watch out. Volume is measured in cubic units, so the answer ends with a small 3, such as cm³.
Method. Multiply the length, width and height.
- Write the formula. Volume of a cuboid = length x width x height.3 x 3 x 3
- Multiply. Multiply the three lengths.3 x 3 = 9, then 9 x 3 = 27
- Write the units. Cubic units for volume.27 cm³
- Write the answer. 27 cm³
Check. 3 x 3 x 3 = 27. ✓
Watch out. Volume is measured in cubic units, so the answer ends with a small 3, such as cm³.
Method. Multiply the length, width and height.
- Write the formula. Volume of a cuboid = length x width x height.10 x 4 x 2
- Multiply. Multiply the three lengths.10 x 4 = 40, then 40 x 2 = 80
- Write the units. Cubic units for volume.80 cm³
- Write the answer. 80 cm³
Check. 10 x 4 x 2 = 80. ✓
Watch out. Volume is measured in cubic units, so the answer ends with a small 3, such as cm³.
Method. Work out both volumes.
- Cube. 5 x 5 x 5 = 125 cm3.125
- Cuboid. 4 x 6 x 4 = 96 cm3.96
- Compare. 125 is more than 96, so the cube holds more.5 cm cube
- Write the answer. 5 cm cube
Check. 125 > 96. ✓
Method. Divide the volume by the length and the width.
- Area of the base. 6 x 4 = 24 cm2.24
- Height. 48 / 24 = 2 cm.2
- Write the answer. 2
Check. 24 x 2 = 48. ✓
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. 3 + 4 = 7, and the perimeter is 14 cm, which is the smallest of the three.3 cm by 4 cm fits
- Reject the other choices. 2 by 6 has a perimeter of 16 and 1 by 12 has a perimeter of 26.2 cm by 6 cm and 1 cm by 12 cm do not fit
- Write the answer. 3 cm by 4 cm
Check. 3 cm by 4 cm is the one that fits. ✓
Method. Work out both perimeters.
- Perimeter of 6 by 2. 2 x (6 + 2) = 16 cm.16 cm
- Perimeter of 4 by 3. 2 x (4 + 3) = 14 cm.14 cm
- Write the answer. Because the perimeter depends on the total length of the sides, and 6 + 2 = 8 is not the same as 4 + 3 = 7.
Check. 16 and 14 are different. ✓
Method. The number of cubes is the same as the volume.
- Volume. 10 x 5 x 2 = 100 cm3.100
- Count. Each 1 cm cube is 1 cm3, so 100 cubes fill it.100
- Write the answer. 100
Check. 100 / (5 x 2) = 10. ✓
Method. List the pairs that multiply to the area, then find each perimeter.
- List the whole-number pairs that multiply to the area. Each pair of sides gives area 16.1 x 16; 2 x 8; 4 x 4
- Work out each perimeter. Perimeter = 2 x (length + width).1 by 16 gives 34; 2 by 8 gives 20; 4 by 4 gives 16
- Choose the smallest perimeter. The smallest perimeter is the one with the sides closest in length.4 by 4 (perimeter 16)
- Write the answer. 4 by 4 (perimeter 16)
Check. The closest pair of sides gives the smallest perimeter. ✓
Watch out. The same area can have very different perimeters: a long thin rectangle has a bigger perimeter than a square.
Method. Find the width from the area, then the perimeter.
- Width. 30 / 6 = 5 cm.5 cm
- Perimeter. 2 x (6 + 5) = 22 cm.22 cm
- Write the answer. 5 cm and 22 cm
Check. 6 x 5 = 30 and 2 x 11 = 22. ✓
Method. Work out both perimeters.
- 25 by 1. 2 x (25 + 1) = 52 m.52 m
- 5 by 5. 2 x (5 + 5) = 20 m.20 m
- Write the answer. 5 m by 5 m, because its perimeter is 20 m, which is less than the 52 m of 25 by 1.
Check. 20 is less than 52. ✓