Perimeter and Area: How to Tackle Compound Shapes in the 11+ Maths Exam

Why Compound Shapes Trip Children Up

Most children learn the formula for the area of a rectangle very early, and they can usually recite that the perimeter is “all the sides added together”. Then the 11+ paper presents a shape that is a rectangle with a semicircle stuck on one end, labels only three of the sides, and the confidence evaporates.

The mathematics involved has not changed at all. What has changed is that the child must now do two things before they touch a formula: break the shape into parts they recognise, and work out the lengths the diagram has not given them. Master those two habits and compound shape questions become some of the most reliable marks on the paper.

Did you know? Compound shape questions are a favourite with examiners precisely because they test several skills at once — formula recall, subtraction, spotting equal sides, and careful reading. One question can carry three or four marks, so it is well worth the practice time.

The Formulas Your Child Must Know by Heart

Before tackling anything compound, these need to be automatic. If your child has to pause and think, they are not ready for the harder questions yet.

Two points cause more lost marks than anything else here:

• The radius is the distance from the centre of the circle to the edge. The diameter goes all the way across and is twice the radius. Questions almost always give you one and expect you to find the other.
• Area answers must carry squared units (cm²), perimeter answers must not (cm). Examiners do take the mark.

A Worked Example: Rectangle Plus Semicircle

Imagine a shape made of a rectangle with a semicircle attached to one end. The diagram labels the total length across the bottom as 16 cm, the height of the rectangle as 7 cm, and the flat section of the base before the curve begins as 6 cm.

Step 1: Find the Missing Lengths

The total base is 16 cm and the flat part is 6 cm, so the remaining stretch — the part the semicircle spans — is:

16 − 6 = 10 cm

That 10 cm is the diameter of the circle, not the radius. Halve it:

radius = 10 ÷ 2 = 5 cm

Because opposite sides of a rectangle are equal, the height on the far side is also 7 cm. The diagram never said so, but the child must know it.

Tip: Get your child into the habit of writing every length they work out straight onto the diagram. A shape with all its sides filled in is a far easier thing to add up than a shape plus a page of loose workings.

Step 2: Calculate the Curved Edge

The full circumference would be:

2 x 3.14 x 5 = 31.4 cm

But only half the circle forms part of this outline, so halve it:

31.4 ÷ 2 = 15.7 cm

Step 3: Add Up the Perimeter

Now trace right around the outside and add every edge you meet:

6 + 7 + 16 + 7 + 15.7 = 51.7 cm

Notice what is not in that sum: the flat diameter where the semicircle meets the rectangle. It is an internal line, not part of the outside edge. Including it is the single most common error in this question type.

Step 4: Split the Shape for Area

Area needs a different approach. Divide the shape into the parts you recognise — here, a rectangle and a semicircle — and work each one out separately.

Rectangle: 7 x 16 = 112 cm²

Semicircle: (3.14 x 5²) ÷ 2 = (3.14 x 25) ÷ 2 = 78.5 ÷ 2 = 39.25 cm²

Step 5: Add the Parts Together

112 + 39.25 = 151.25 cm²

Two answers, from one diagram: a perimeter of 51.7 cm and an area of 151.25 cm².

The Method in Five Steps

Whatever the compound shape, the routine is identical. Teach it once and your child can apply it to any variation the examiner invents.

Tip: The finger-trace in step 3 sounds childish, but it works. If your finger never travels along a line, that line does not belong in the perimeter. Encourage your child to highlight the outline in pencil in the exam.

Common Traps to Watch For

• Using the diameter as the radius. If the question gives you the distance all the way across, halve it before it goes anywhere near a formula.
• Forgetting to halve a semicircle. Both the circumference and the area of a half circle need dividing by two. Children frequently remember one and forget the other.
• Adding the internal line to the perimeter. Covered above — but worth repeating, because it is the trap examiners rely on.
• Subtracted shapes. Sometimes a shape has a piece cut out rather than added on. The area is then the big shape minus the missing piece, but the perimeter still follows the outside edge, which may now include a curve where the hole meets the boundary.
• Mixed units. If one side is given in millimetres and another in centimetres, convert before doing anything else.

Fitting This Into Wider Maths Preparation

Perimeter and area sit inside the broader “shape, space and measures” strand of the 11+, and they lean heavily on skills from elsewhere on the paper. Confident multiplication, accurate decimal arithmetic and clean subtraction are all prerequisites — which is why our guide on how to improve your child’s 11+ maths score puts number fluency first. If decimals such as 39.25 and 15.7 still cause hesitation, spending a fortnight on fractions, decimals and percentages will pay for itself here.

Once the method is secure, the best thing your child can do is meet these questions under timed conditions alongside everything else on the paper. Our collection of 11+ maths test questions and answers is a good place to see how shape questions sit within a full paper.

A Two-Week Practice Plan

Practise Perimeter and Area with Our 11+ Maths Apps

Compound shape questions reward pattern recognition, and pattern recognition comes from volume. Two of our apps are built for exactly this.

11+ Maths Learn & Test is where the method gets learned. It offers 1,280 questions across 32 topics, with detailed study notes covering shapes, algebra, fractions, data handling, time, money and more. The shape topics walk through the formulas and worked examples step by step, so a child who is shaky on radius versus diameter can rebuild the foundation before moving on.

11+ Maths Practice Papers is where the method gets tested. It provides 1,200 maths questions across 24 full practice papers covering all 11+ maths topics, each with detailed explanations. Perimeter and area questions appear interleaved with everything else, exactly as they will on exam day — which is the only way to find out whether your child can recall the right formula under pressure rather than when they already know a shape question is coming.

Used together with the rest of our suite — over 8,100 questions across 7 specialist 11+ apps spanning maths, English, verbal reasoning, non-verbal reasoning and vocabulary — they give your child structured, explained practice across every part of the exam.

Perimeter and area are among the most learnable topics on the 11+ maths paper. There is no trick to them, only a method: split the shape, fill in the lengths, trace the outline, add the parts, check the units. Drill that sequence until it is second nature, and these questions stop being a hurdle and start being the marks your child banks with confidence.