Improve Tuition · Maths · Perimeter & Area · Grade 2–4
Diagnostic practice paper

2D Shapes: Perimeter and Area

Build the skills one section at a time — and find the gap if there is one.

35 questions · 7 skill sections · diagrams & misconception feedback · no time pressure

For the tutor

Where the pupil stops tells you the gap

AHexagon PerimeterIf they slip here: Counting every side of every hexagon, including the hidden joined sides.Grade 2–3
BRectangle Area and PerimeterIf they slip here: Confusing area with perimeter, or giving the area as the answer.Grade 3–4
CCardboard Tunnel AreaIf they slip here: Including the base when the tunnel has no base.Grade 3–4
DCompound Garden AreaIf they slip here: Using a total length with the wrong height, or missing a piece.Grade 3–4
ETriangle AreaIf they slip here: Forgetting to halve — giving base × height (double the answer).Grade 3–4
FArea Left OverIf they slip here: Adding the inner shapes instead of subtracting them.Grade 3–4
GPaint CoverageIf they slip here: Rounding down, or ignoring the leftover area that still needs a tin.Grade 4
Tutor view Diagnostic notes for each section▸
Section A · Hexagon Perimeter

Skill: Count only the outside edges. Watch for: Counting every side of every hexagon, including the hidden joined sides.

Section B · Rectangle Area and Perimeter

Skill: Use factor pairs to find missing side lengths. Watch for: Confusing area with perimeter, or giving the area as the answer.

Section C · Cardboard Tunnel Area

Skill: Add only the exposed faces (top + two sides). No base. Watch for: Including the base when the tunnel has no base.

Section D · Compound Garden Area

Skill: Split an L-shape into rectangles and add them. Watch for: Using a total length with the wrong height, or missing a piece.

Section E · Triangle Area

Skill: Use ½ × base × height. Watch for: Forgetting to halve — giving base × height (double the answer).

Section F · Area Left Over

Skill: Subtract occupied areas from the whole area. Watch for: Adding the inner shapes instead of subtracting them.

Section G · Paint Coverage

Skill: Divide by coverage and round UP. Watch for: Rounding down, or ignoring the leftover area that still needs a tin.

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What this paper checks

This paper builds your perimeter and area skills one step at a time, then shows you exactly where you slip if you do. Each section targets one skill and one common mistake. Work untimed — the goal is understanding, not speed.

The trickiest sections have a diagram, because seeing the shape stops the most common errors. Here are the three that catch most pupils out.

Triangles — why we halve

halfother halfbase = 6h = 4

A triangle is exactly half of the rectangle around it. So area = ½ × base × height. The most common mistake is doing base × height and forgetting to halve — that gives the whole rectangle, double the real answer.

Tunnels — count only the faces that are there

Three painted faces — no baseSIDE12 × 5TOP12 × 5SIDE12 × 5base — open, not counted

An open tunnel has a top and two sides — but no base. Add only the faces that actually exist. Including the base is the classic slip.

Area left over — subtract, don’t add

30 m20 m10×8shaded = removed

To find what is left (grass, parking, open space), find the whole area and subtract the shapes inside it. Adding the inner shapes is the trap.

How to read the diagrams: a solid coral outline is a face or shape you count; a dashed grey outline is something you do not count; gold hatching marks an area being removed. Every cue is also written in words, so the diagrams work in black and white too.

Choose how to work
Section A · Hexagon Perimeter
Count only the outside edges.
Learning objective 1 · Grade 2–3

Find a perimeter by counting only the outside edges.

Success criteria — I can:
  • know perimeter is the total distance around the outside
  • count only the edges on the outside boundary
  • multiply the number of outside edges by the edge length
Q1.A patio is made from regular hexagonal slabs. Each side is 20 cm long. The outside edge of the patio has 16 sides. What is the perimeter?
This patio has 16 outside edges (coral). The joined edges between slabs (✗) are hidden, so they are not counted.
✗✗✗✗
outside edge — counts ✗ joined edge — hidden
Your answer:(1 mark)
  1. Perimeter = number of outside edges × edge length.
  2. 16 × 20 = 320 cm.
Q2.A garden path is made from regular hexagonal tiles. Each side is 25 cm long. The outside edge has 18 sides. What is the perimeter?
This path has 18 outside edges (coral). The joined edges between slabs (✗) are hidden, so they are not counted.
✗✗✗
outside edge — counts ✗ joined edge — hidden
Your answer:(1 mark)
  1. 18 × 25 = 450 cm.
Q3.A floor pattern is made from regular hexagons. Each side is 15 cm long. The outside perimeter has 22 sides. What is the perimeter?
This floor pattern has 22 outside edges (coral). The joined edges between slabs (✗) are hidden, so they are not counted.
✗✗✗✗
outside edge — counts ✗ joined edge — hidden
Your answer:(1 mark)
  1. 22 × 15 = 330 cm.
Q4.A patio is made from regular hexagonal slabs. Each side is 40 cm long. There are 14 outside edges. What is the perimeter?
This patio has 14 outside edges (coral). The joined edges between slabs (✗) are hidden, so they are not counted.
✗✗
outside edge — counts ✗ joined edge — hidden
Your answer:(1 mark)
  1. 14 × 40 = 560 cm.
Q5.A honeycomb design is made from regular hexagons. Each side is 30 cm long. The outside boundary has 20 sides. What is the perimeter?
This honeycomb design has 20 outside edges (coral). The joined edges between slabs (✗) are hidden, so they are not counted.
✗✗✗✗✗
outside edge — counts ✗ joined edge — hidden
Your answer:(1 mark)
  1. 20 × 30 = 600 cm.
Section B · Rectangle Area and Perimeter
Use factor pairs to find missing side lengths.
Learning objective 2 · Grade 3–4

Find missing side lengths from an area, then the perimeter.

Success criteria — I can:
  • use factor pairs of the area to find the two side lengths
  • pick the pair with the correct difference
  • find perimeter as 2 × (length + width)
Q6.The length of a rectangle is 4 cm longer than its width. The area is 32 cm². The sides are whole numbers. What is the perimeter?
Area fills the inside (given); perimeter is the distance around the outside (what you want). Find the two side lengths from the area, then add all four sides.
Area = 32 cm²(this is given)width = ?length = width + 4Perimeter = the outline (what to find)
Your answer:(1 mark)
  1. Find two numbers that multiply to 32 with a difference of 4: 4 and 8.
  2. Perimeter = 2 × (4 + 8) = 24 cm.
Q7.The length is 3 cm longer than the width. The area is 40 cm². The sides are whole numbers. What is the perimeter?
Area fills the inside (given); perimeter is the distance around the outside (what you want). Find the two side lengths from the area, then add all four sides.
Area = 40 cm²(this is given)width = ?length = width + 3Perimeter = the outline (what to find)
Your answer:(1 mark)
  1. 5 × 8 = 40, difference 3.
  2. 2 × (5 + 8) = 26 cm.
Q8.The length is 6 cm longer than the width. The area is 72 cm². The sides are whole numbers. What is the perimeter?
Area fills the inside (given); perimeter is the distance around the outside (what you want). Find the two side lengths from the area, then add all four sides.
Area = 72 cm²(this is given)width = ?length = width + 6Perimeter = the outline (what to find)
Your answer:(1 mark)
  1. 6 × 12 = 72, difference 6.
  2. 2 × (6 + 12) = 36 cm.
Q9.The length is 5 cm longer than the width. The area is 50 cm². The sides are whole numbers. What is the perimeter?
Area fills the inside (given); perimeter is the distance around the outside (what you want). Find the two side lengths from the area, then add all four sides.
Area = 50 cm²(this is given)width = ?length = width + 5Perimeter = the outline (what to find)
Your answer:(1 mark)
  1. 5 × 10 = 50, difference 5.
  2. 2 × (5 + 10) = 30 cm.
Q10.The length is 7 cm longer than the width. The area is 78 cm². The sides are whole numbers. What is the perimeter?
Area fills the inside (given); perimeter is the distance around the outside (what you want). Find the two side lengths from the area, then add all four sides.
Area = 78 cm²(this is given)width = ?length = width + 7Perimeter = the outline (what to find)
Your answer:(1 mark)
  1. 6 × 13 = 78, difference 7.
  2. 2 × (6 + 13) = 38 cm.
Section C · Cardboard Tunnel Area
Add only the exposed faces (top + two sides). No base.
Learning objective 3 · Grade 3–4

Add only the exposed faces of an open solid.

Success criteria — I can:
  • identify which faces are actually there (top and two sides)
  • find the area of each face
  • leave out the base because the tunnel is open underneath
Q11.A cardboard tunnel has length 12 cm, width 5 cm, height 5 cm. It has 3 outside faces: top, left side and right side. What is the total outside area?
Count only the faces that are there: top + two sides (no base).
Three painted faces — no baseSIDE12 × 5TOP12 × 5SIDE12 × 5base — open, not counted
Your answer:(1 mark)
  1. Top = length × width = 12 × 5 = 60 cm².
  2. Two sides = length × height × 2 = 12 × 5 × 2 = 120 cm².
  3. Total = 60 + 120 = 180 cm².
Q12.A toy car tunnel has length 18 cm, width 7 cm, height 7 cm. Find the total outside area (top and two sides only).
Count only the faces that are there: top + two sides (no base).
Three painted faces — no baseSIDE18 × 7TOP18 × 7SIDE18 × 7base — open, not counted
Your answer:(1 mark)
  1. Top = 18 × 7 = 126 cm².
  2. Two sides = 18 × 7 × 2 = 252 cm².
  3. Total = 378 cm².
Q13.A cardboard tunnel has length 20 cm, width 8 cm, height 6 cm. Find the total outside area (top and two sides only).
Count only the faces that are there: top + two sides (no base).
Three painted faces — no baseSIDE20 × 6TOP20 × 8SIDE20 × 6base — open, not counted
Your answer:(1 mark)
  1. Top = 20 × 8 = 160 cm².
  2. Two sides = 20 × 6 × 2 = 240 cm².
  3. Total = 400 cm².
Q14.A tunnel is made from cardboard: length 25 cm, width 10 cm, height 8 cm. Only the outside top and two sides are covered. What is the total outside area?
Count only the faces that are there: top + two sides (no base).
Three painted faces — no baseSIDE25 × 8TOP25 × 10SIDE25 × 8base — open, not counted
Your answer:(1 mark)
  1. Top = 25 × 10 = 250 cm².
  2. Two sides = 25 × 8 × 2 = 400 cm².
  3. Total = 650 cm².
Q15.A cardboard tunnel has length 30 cm, width 12 cm, height 9 cm. The tunnel has no base. Find the total outside area.
Count only the faces that are there: top + two sides (no base).
Three painted faces — no baseSIDE30 × 9TOP30 × 12SIDE30 × 9base — open, not counted
Your answer:(1 mark)
  1. Top = 30 × 12 = 360 cm².
  2. Two sides = 30 × 9 × 2 = 540 cm².
  3. Total = 900 cm².
Section D · Compound Garden Area
Split an L-shape into rectangles and add them.
Learning objective 4 · Grade 3–4

Split a compound (L-shaped) area into rectangles.

Success criteria — I can:
  • split the shape into two rectangles
  • find each missing side length
  • find each rectangle’s area and add them
Q16.A garden is made from 2 rectangles. Left: width 4 m, height 7 m. Right: width 3 m, height 5 m. What is the total area?
Split the shape into two rectangles, find each area, then add them together.
14 m7 m23 m5 m
Your answer:(1 mark)
  1. Left = 4 × 7 = 28 m².
  2. Right = 3 × 5 = 15 m².
  3. Total = 28 + 15 = 43 m².
Q17.A garden is made from 2 rectangles. Left: width 6 m, height 8 m. Right: width 4 m, height 3 m. What is the total area?
Split the shape into two rectangles, find each area, then add them together.
16 m8 m24 m3 m
Your answer:(1 mark)
  1. Left = 6 × 8 = 48 m².
  2. Right = 4 × 3 = 12 m².
  3. Total = 60 m².
Q18.An L-shaped garden has a tall left part (width 5 m, height 10 m) and a shorter right part (width 4 m, height 6 m). Calculate the total area.
Split the shape into two rectangles, find each area, then add them together.
15 m10 m24 m6 m
Your answer:(1 mark)
  1. Left = 5 × 10 = 50 m².
  2. Right = 4 × 6 = 24 m².
  3. Total = 74 m².
Q19.An L-shaped garden has a total bottom length of 12 m. Left rectangle: width 7 m, height 9 m. Right rectangle height = 4 m. Work out the total area.
Split into two rectangles. First find the missing width: total bottom − the known width. Then find each area and add them.
17 m9 m2?4 mtotal bottom = 12 m
Your answer:(1 mark)
  1. Right width = 12 − 7 = 5 m.
  2. Left = 7 × 9 = 63 m².
  3. Right = 5 × 4 = 20 m².
  4. Total = 83 m².
Q20.An L-shaped garden: total bottom length 15 m, left width 6 m, left height 11 m, right height 5 m. Calculate the total area.
Split into two rectangles. First find the missing width: total bottom − the known width. Then find each area and add them.
16 m11 m2?5 mtotal bottom = 15 m
Your answer:(1 mark)
  1. Right width = 15 − 6 = 9 m.
  2. Left = 6 × 11 = 66 m².
  3. Right = 9 × 5 = 45 m².
  4. Total = 111 m².
Section E · Triangle Area
Use ½ × base × height.
Learning objective 5 · Grade 3–4

Find the area of a triangle using ½ × base × height.

Success criteria — I can:
  • know a triangle is half of its surrounding rectangle
  • multiply base × height
  • then halve the result
Q21.A triangular part of a garden floods. Base = 6 m, height = 4 m. What area floods?
A triangle is exactly half of the rectangle around it.
halfother halfbase = 6h = 4
Your answer:(1 mark)
  1. ½ × base × height = ½ × 6 × 4.
  2. = ½ × 24 = 12 m².
Q22.A triangular area in a park floods. Base = 10 m, height = 5 m. Calculate the flooded area.
A triangle is exactly half of the rectangle around it.
halfother halfbase = 10h = 5
Your answer:(1 mark)
  1. ½ × 10 × 5 = 25 m².
Q23.A triangular section of a garden floods. Base = 12 m, height = 7 m. What is the flooded area?
A triangle is exactly half of the rectangle around it.
halfother halfbase = 12h = 7
Your answer:(1 mark)
  1. ½ × 12 × 7 = 42 m².
Q24.A garden has a triangular flooded area. Base = 14 m, height = 9 m. Calculate the area that floods.
A triangle is exactly half of the rectangle around it.
halfother halfbase = 14h = 9
Your answer:(1 mark)
  1. ½ × 14 × 9 = 63 m².
Q25.A triangular area floods in a garden. Base = 18 m, height = 11 m. What is the flooded area?
A triangle is exactly half of the rectangle around it.
halfother halfbase = 18h = 11
Your answer:(1 mark)
  1. ½ × 18 × 11 = 99 m².
Section F · Area Left Over
Subtract occupied areas from the whole area.
Learning objective 6 · Grade 3–4

Subtract the occupied areas from the whole area.

Success criteria — I can:
  • find the area of the whole rectangle
  • find the area of each inner shape
  • subtract every inner area from the whole
Q26.A rectangular playground is 30 m long and 20 m wide. A sandpit inside is 10 m by 8 m. The rest is grass. What is the area of the grass?
Find the whole area, then subtract the shapes inside it.
30 m20 m10×8shaded = removed
Your answer:(1 mark)
  1. Whole = 30 × 20 = 600 m².
  2. Sandpit = 10 × 8 = 80 m².
  3. Grass = 600 − 80 = 520 m².
Q27.A rectangular field is 50 m long and 30 m wide. A tennis court inside is 20 m by 12 m. The rest is open space. What is the area of the open space?
Find the whole area, then subtract the shapes inside it.
50 m30 m20×12shaded = removed
Your answer:(1 mark)
  1. Whole = 50 × 30 = 1500 m².
  2. Court = 20 × 12 = 240 m².
  3. Open space = 1260 m².
Q28.A rectangular car park is 60 m long and 40 m wide. A shop building takes up 25 m by 18 m. The rest is parking space. What is the parking area?
Find the whole area, then subtract the shapes inside it.
60 m40 m25×18shaded = removed
Your answer:(1 mark)
  1. Whole = 60 × 40 = 2400 m².
  2. Shop = 25 × 18 = 450 m².
  3. Parking = 1950 m².
Q29.A rectangular garden is 45 m long and 28 m wide. A pond is 16 m by 9 m and a shed is 8 m by 5 m. What is the area left over?
Find the whole area, then subtract the shapes inside it.
45 m28 m16×98×5shaded = removed
Your answer:(1 mark)
  1. Whole = 45 × 28 = 1260 m².
  2. Pond = 144 m², Shed = 40 m².
  3. Left over = 1260 − 144 − 40 = 1076 m².
Q30.A rectangular shopping site is 80 m long and 50 m wide. A supermarket is 35 m by 22 m and a storage area is 18 m by 10 m. The rest is car park. What is the car park area?
Find the whole area, then subtract the shapes inside it.
80 m50 m35×2218×10shaded = removed
Your answer:(1 mark)
  1. Whole = 80 × 50 = 4000 m².
  2. Supermarket = 770 m², Storage = 180 m².
  3. Car park = 4000 − 770 − 180 = 3050 m².
Section G · Paint Coverage
Divide by coverage and round UP.
Learning objective 7 · Grade 4

Divide total area by coverage and round up to whole tins.

Success criteria — I can:
  • find the total area to be painted
  • divide by the area one tin covers
  • round UP — a part-tin still means buying a whole tin
Q31.A room has 4 walls, each 3 m wide and 2 m tall. One tin covers 10 m². How many tins are needed?
Add up the area of every wall, divide by what one tin covers, then round UP — a part-tin still means buying a whole tin.
Paint every wall — find the total area3×23×23×23×21 tin covers 10 m²
Your answer:(1 mark)
  1. Wall area = 4 × (3 × 2) = 24 m².
  2. 24 ÷ 10 = 2.4 tins.
  3. Round UP: 3 tins.
Q32.Mariam paints 4 walls, each 5 m wide and 2 m tall. One tin covers 12 m². How many tins does she need?
Add up the area of every wall, divide by what one tin covers, then round UP — a part-tin still means buying a whole tin.
Paint every wall — find the total area5×25×25×25×21 tin covers 12 m²
Your answer:(1 mark)
  1. Area = 4 × (5 × 2) = 40 m².
  2. 40 ÷ 12 = 3.33.
  3. Round UP: 4 tins.
Q33.Ali paints 3 walls, each 4 m wide and 3 m tall. One tin covers 15 m². How many tins does Ali need?
Add up the area of every wall, divide by what one tin covers, then round UP — a part-tin still means buying a whole tin.
Paint every wall — find the total area4×34×34×31 tin covers 15 m²
Your answer:(1 mark)
  1. Area = 3 × (4 × 3) = 36 m².
  2. 36 ÷ 15 = 2.4.
  3. Round UP: 3 tins.
Q34.A classroom has 4 walls, each 6 m wide and 3 m tall. One tin covers 20 m². How many tins are needed?
Add up the area of every wall, divide by what one tin covers, then round UP — a part-tin still means buying a whole tin.
Paint every wall — find the total area6×36×36×36×31 tin covers 20 m²
Your answer:(1 mark)
  1. Area = 4 × (6 × 3) = 72 m².
  2. 72 ÷ 20 = 3.6.
  3. Round UP: 4 tins.
Q35.A hall has 4 walls. Two are 8 m wide and 3 m tall; the other two are 5 m wide and 3 m tall. One tin covers 18 m². How many tins are needed?
Add up the area of every wall, divide by what one tin covers, then round UP — a part-tin still means buying a whole tin.
Paint every wall — find the total area8×38×35×35×31 tin covers 18 m²
Your answer:(1 mark)
  1. Two walls = 8 × 3 × 2 = 48 m². Two walls = 5 × 3 × 2 = 30 m².
  2. Total = 78 m². 78 ÷ 18 = 4.33.
  3. Round UP: 5 tins.