For: Year 5 and Year 6 | Exams: KS2 SATs and 11+ | Calculator: not allowed
Jump to: Level 1 – SATs core · Level 2 – SATs stretch · Level 3 – 11+ standard · Level 4 – 11+ selective
📌 KEY FACTS TO KNOW BY HEART
Perimeter: the distance all the way round the edge. Rectangle: 2 × (length + width). Units: cm, m.
Area: rectangle = length × width · triangle = ½ × base × perpendicular height · parallelogram = base × perpendicular height. Units: cm², m².
Volume: cuboid = length × width × height. Units: cm³, m³.
Units: 1 m² = 10000 cm² · 1 cm³ = 1 ml · 1000 cm³ = 1 litre
Level 1 – SATs core: perimeter, area and volume of rectangles and cuboids
Matches KS2 SATs Paper 1 and the easier reasoning questions. Always write the units.
✏️ WORKED EXAMPLE: A rectangle is 8 cm long and 5 cm wide. Find its perimeter and its area.
Step 1: perimeter: add all four sides: 8 + 5 + 8 + 5 = 26 cm
Step 2: area: length × width = 8 × 5 = 40 cm²
Answer: perimeter 26 cm, area 40 cm²
Common mistake: adding only two sides (8 + 5 = 13 cm) for the perimeter, or writing cm instead of cm² for the area.
Your turn
Q1.1 A rectangle is 7 cm long and 3 cm wide. What is its perimeter?
Show answer
7 + 3 + 7 + 3 = 20 cm
Q1.2 A rectangle is 9 cm long and 4 cm wide. What is its area?
Show answer
9 × 4 = 36 cm²
Q1.3 A square has sides of 11 cm. What is its area?
Show answer
11 × 11 = 121 cm²
Q1.4 A square has a perimeter of 28 cm. How long is each side?
Show answer
A square has 4 equal sides: 28 ÷ 4 = 7 cm
Q1.5 A rectangular playground is 12 m long and 5 m wide. What is its area?
Show answer
12 × 5 = 60 m²
Q1.6 A cuboid is built from centimetre cubes. It is 4 cubes long, 3 cubes wide and 2 cubes tall. How many cubes are used?
Show answer
One layer has 4 × 3 = 12 cubes. Two layers: 12 × 2 = 24 cubes (a volume of 24 cm³)
Q1.7 A cuboid measures 5 cm by 4 cm by 3 cm. What is its volume?
Show answer
5 × 4 × 3 = 60 cm³
Q1.8 A cube has edges 3 cm long. What is its volume?
Show answer
3 × 3 × 3 = 27 cm³
Q1.9 A rectangular garden is 15 m long and 8 m wide. A fence goes all the way round it. How long is the fence?
Show answer
2 × (15 + 8) = 2 × 23 = 46 m
Q1.10 A rectangular rug is 2.5 m long and 2 m wide. What is its area?
Show answer
2.5 × 2 = 5 m²
Level 2 – SATs stretch: triangles, parallelograms, missing sides and units
Matches the harder questions in SATs Papers 2 and 3 (reasoning).
✏️ WORKED EXAMPLE: A triangle has a base of 10 cm and a perpendicular height of 7 cm. What is its area?
Step 1: a rectangle with the same base and height has area 10 × 7 = 70 cm²
Step 2: the triangle is exactly half of that rectangle: 70 ÷ 2 = 35 cm²
Common mistake: forgetting to halve and giving 70 cm², or using a sloping side instead of the perpendicular (straight up) height.
Your turn
Q2.1 A triangle has a base of 8 cm and a perpendicular height of 5 cm. What is its area?
Show answer
8 × 5 = 40, then halve: 20 cm²
Q2.2 A parallelogram has a base of 9 cm and a perpendicular height of 4 cm. Its sloping sides are 5 cm long. What is its area?
Show answer
Base × perpendicular height = 9 × 4 = 36 cm². The 5 cm sloping side is not needed.
Q2.3 A rectangle has an area of 56 cm². One side is 7 cm long. How long is the other side?
Show answer
56 ÷ 7 = 8 cm
Q2.4 A rectangle has a perimeter of 30 cm. Its length is 9 cm. What is its width?
Show answer
Length + width = 30 ÷ 2 = 15 cm, so width = 15 − 9 = 6 cm
Q2.5 How many cm³ are there in 2.5 litres?
Show answer
1 litre = 1000 cm³, so 2.5 × 1000 = 2500 cm³
Q2.6 A right-angled triangle has two sides of 6 cm and 9 cm that meet at the right angle. What is its area?
Show answer
These two sides are the base and the height: 6 × 9 = 54, then halve: 27 cm²
Q2.7 Write 3 m² in cm².
Show answer
1 m² = 100 cm × 100 cm = 10000 cm², so 3 m² = 30000 cm²
Q2.8 A fish tank is a cuboid 40 cm long, 25 cm wide and 30 cm high. How many litres does it hold when it is full?
Show answer
40 × 25 × 30 = 30000 cm³. 1000 cm³ = 1 litre, so it holds 30 litres
Q2.9 A triangle has an area of 24 cm² and a base of 8 cm. What is its perpendicular height?
Show answer
Base × height = 2 × 24 = 48, so height = 48 ÷ 8 = 6 cm
Q2.10 A cuboid has a volume of 120 cm³. It is 6 cm long and 5 cm wide. How tall is it?
Show answer
Area of base = 6 × 5 = 30 cm². Height = 120 ÷ 30 = 4 cm
Level 3 – 11+ standard: compound shapes, comparing shapes and filling tanks
Typical of grammar-school 11+ papers. Split shapes into rectangles, and check which measurement the question wants.
✏️ WORKED EXAMPLE: A rectangle 10 cm long and 8 cm wide has a smaller rectangle, 4 cm long and 3 cm wide, cut from one corner to make an L-shape. Find the area and the perimeter of the L-shape.
Step 1: area of the big rectangle = 10 × 8 = 80 cm²
Step 2: area cut away = 4 × 3 = 12 cm², so the L-shape has area 80 − 12 = 68 cm²
Step 3: go round the edge: 10 + 5 + 4 + 3 + 6 + 8 = 36 cm
Answer: area 68 cm², perimeter 36 cm
Common mistake: thinking the perimeter gets smaller. The two new edges (4 cm and 3 cm) exactly replace the pieces of edge that were cut off, so the perimeter stays the same as the big rectangle.
Your turn
Q3.1 An L-shape is made by joining two rectangles edge to edge, without overlapping. One rectangle is 6 cm by 2 cm and the other is 4 cm by 3 cm. What is the area of the L-shape?
Show answer
6 × 2 = 12 and 4 × 3 = 12, so the total is 24 cm²
Q3.2 A rectangular lawn is 12 m long and 9 m wide. A square flower bed with sides of 3 m is dug in the middle of the lawn. What area of grass is left?
Show answer
12 × 9 = 108 m². Flower bed: 3 × 3 = 9 m². Grass: 108 − 9 = 99 m²
Q3.3 A rectangle is 6 cm by 4 cm. A square has sides of 5 cm. They have the same perimeter. Which shape has the larger area, and by how much?
Show answer
Both perimeters are 20 cm. Rectangle area = 24 cm², square area = 25 cm². The square, by 1 cm²
Q3.4 A rectangle has an area of 36 cm². Its sides are whole numbers of centimetres. What is the largest perimeter it could have?
Show answer
The possible sides are 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6. The longest, thinnest one, 1 cm by 36 cm, has perimeter 2 × 37 = 74 cm
Q3.5 A path 1 m wide goes all the way round the outside of a rectangular pond that is 6 m long and 4 m wide. What is the area of the path?
Show answer
The pond and path together make a rectangle 8 m by 6 m = 48 m². Subtract the pond: 48 − 24 = 24 m²
Q3.6 A tank is a cuboid 50 cm long, 40 cm wide and 30 cm high. Water is poured in until it is 20 cm deep. How many litres of water are in the tank?
Show answer
50 × 40 × 20 = 40000 cm³ = 40 litres
Q3.7 The floor of a room is an L-shape with six straight sides that meet at right angles. Starting at one corner and going round, the sides are 8 m, 4 m, 3 m, 2 m, 5 m and 6 m long, in that order. What is the area of the floor?
Show answer
The longest sides, 8 m and 6 m, make a rectangle 8 × 6 = 48 m². The missing corner is 3 m by 2 m = 6 m². Area = 48 − 6 = 42 m²
Q3.8 A rectangle has a perimeter of 24 cm. Its length is twice its width. What is its area?
Show answer
Going round: width + length + width + length = 6 widths = 24 cm, so the width is 4 cm and the length is 8 cm. Area = 8 × 4 = 32 cm²
Q3.9 A tank is a cuboid 30 cm long, 20 cm wide and 25 cm high. A jug holds 1.5 litres. How many full jugs of water are needed to fill the tank?
Show answer
30 × 20 × 25 = 15000 cm³ = 15 litres. 15 ÷ 1.5 = 10 jugs
Q3.10 A square has the same area as a rectangle that is 16 cm long and 4 cm wide. What is the difference between their perimeters?
Show answer
Area = 64 cm², so the square has sides of 8 cm and perimeter 32 cm. The rectangle has perimeter 2 × 20 = 40 cm. Difference: 8 cm
Level 4 – 11+ selective: cubes, reverse problems and scaling
The style of selective independent-school papers. Think about how lengths, areas and volumes are linked.
✏️ WORKED EXAMPLE: A rectangle is 3 cm long and 2 cm wide. Both its length and its width are doubled. How many times bigger is its area?
Step 1: old area = 3 × 2 = 6 cm²
Step 2: new rectangle is 6 cm by 4 cm, so new area = 24 cm²
Step 3: 24 ÷ 6 = 4, so the area is 4 times bigger
Check: four copies of the old 3 cm by 2 cm rectangle fit exactly inside the new one ✓
Common mistake: saying the area doubles. Doubling both lengths multiplies the area by 2 × 2 = 4 (and the volume of a cuboid by 2 × 2 × 2 = 8).
Your turn
Q4.1 A box is a cuboid 20 cm long, 10 cm wide and 10 cm high. How many cubes with 5 cm edges fit exactly inside it?
Show answer
20 ÷ 5 = 4, 10 ÷ 5 = 2, 10 ÷ 5 = 2. So 4 × 2 × 2 = 16 cubes
Q4.2 A square has an area of 49 cm². Each of its sides is made three times as long. What is the area of the new square?
Show answer
Old side = 7 cm, new side = 21 cm. Area = 21 × 21 = 441 cm² (9 times the old area)
Q4.3 A rectangle has an area of 24 cm² and a perimeter of 22 cm. What are its length and width?
Show answer
Length + width = 22 ÷ 2 = 11 cm. Which two numbers add to 11 and multiply to 24? 8 cm and 3 cm
Q4.4 A cube has a volume of 125 cm³. What is the total area of its six faces?
Show answer
5 × 5 × 5 = 125, so each edge is 5 cm. Each face is 25 cm², so 6 × 25 = 150 cm²
Q4.5 A cuboid tank has a base 40 cm long and 25 cm wide. 5 litres of water are poured in. By how many centimetres does the water level rise?
Show answer
5 litres = 5000 cm³. Base area = 40 × 25 = 1000 cm². Rise = 5000 ÷ 1000 = 5 cm
Q4.6 A crate is a cuboid 60 cm long, 40 cm wide and 30 cm high. What is the greatest number of boxes measuring 20 cm by 10 cm by 15 cm that can be packed inside it?
Show answer
Crate volume ÷ box volume = 72000 ÷ 3000 = 24, so no more than 24 can fit. Packing 3 along (60 ÷ 20), 4 across (40 ÷ 10) and 2 up (30 ÷ 15) fits exactly 24 boxes
Q4.7 A photo is 12 cm long and 8 cm wide. It is enlarged so that each side is 1.5 times as long. By how many cm² does its area increase?
Show answer
New photo: 18 cm by 12 cm = 216 cm². Old area: 96 cm². Increase: 216 − 96 = 120 cm²
Q4.8 A large cube is made from 64 small cubes, 4 along each edge. The outside of the large cube is painted. How many small cubes have paint on exactly one face?
Show answer
These are in the middle of each face, not on an edge. Each face has a 2 by 2 middle = 4 cubes. 6 faces × 4 = 24 cubes
Q4.9 A square piece of card has a perimeter of 36 cm. A 2 cm by 2 cm square is cut from each corner, and the sides are folded up to make an open box. What is the volume of the box?
Show answer
The card is 9 cm by 9 cm. The base is 9 − 2 − 2 = 5 cm each way and the box is 2 cm tall. Volume = 5 × 5 × 2 = 50 cm³
Q4.10 A cuboid 5 cm long, 4 cm wide and 3 cm high is built from centimetre cubes. The outside of the cuboid is painted. How many small cubes have paint on exactly two faces?
Show answer
These lie along the 12 edges but are not corners. The 4 edges of length 5 have 3 each (12), the 4 edges of length 4 have 2 each (8) and the 4 edges of length 3 have 1 each (4). Total: 12 + 8 + 4 = 24 cubes