Perimeter and Area of Compound Shapes — KS3 Maths

A compound shape is made by joining two or more simple shapes together. To find the area, split the compound shape into rectangles or triangles, find each area separately, then add (or subtract) them. The perimeter is the total distance around the outside edge only.

What is a compound shape?

A compound shape (also called a composite shape or rectilinear shape) is any shape that is not a standard rectangle, triangle, or circle. Common examples include L-shapes, T-shapes, and shapes with a rectangular notch cut out.

Two strategies for area:

Split and add — divide the shape into simpler pieces and add their areas.

Large rectangle minus cutout — find the area of the enclosing rectangle, then subtract the missing piece.

Finding missing lengths in a compound shape

Before calculating area, you must find any unlabelled side lengths using the given dimensions.

Key rule: Opposite sides of a rectilinear shape are equal when extended to form a full rectangle.

Worked example 1 — finding a missing length in an L-shape

An L-shape has the following labelled sides (in cm):

Overall width = 10, top step width = 6

Overall height = 8, step height = 3

Missing horizontal length = 10 − 6 = 4 cm

Missing vertical length = 8 − 3 = 5 cm

Always label all missing sides before starting any area or perimeter calculation.

Calculating the perimeter of a compound shape

The perimeter is the total length of all the outer edges only. Do not include any internal dividing lines.

Worked example 2 — perimeter of an L-shape

Outer edges (in cm): 10, 3, 6, 5, 4, 8

Perimeter = 10 + 3 + 6 + 5 + 4 + 8 = 36 cm

A common mistake is forgetting the missing sides — make sure you have labelled them all before summing the perimeter.

Calculating the area by splitting into rectangles

Worked example 3 — area of the same L-shape (split and add)

Rectangle A (top step): 6 × 3 = 18 cm²

Rectangle B (lower section): 10 × 5 = 50 cm²

Total area = 18 + 50 = 68 cm²

Check using the large-rectangle-minus-cutout method:

Full enclosing rectangle: 10 × 8 = 80 cm²

Cutout rectangle: 4 × 5 = 20 cm² (the missing piece top-right)

Area = 80 − 20 = 60 cm²

Wait — there is a discrepancy. Let us re-examine. With the step height being 3 and the lower section height being 5 (total 8): Correct split:

Rectangle A (top): 6 × 3 = 18 cm²; Rectangle B (bottom right): 4 × 8 = 32 cm²? That does not work geometrically. The clearest split for an L-shape with width 10, height 8, step width 6, step height 3:

Horizontal split:

Top rectangle: 6 cm wide × 3 cm tall = 18 cm²

Bottom rectangle: 10 cm wide × 5 cm tall = 50 cm²

Total = 68 cm²

Large rectangle minus cutout:

Enclosing rectangle: 10 × 8 = 80 cm²

Cutout (top-right missing piece): 4 cm wide × 3 cm tall = 12 cm²

Area = 80 − 12 = 68 cm² ✓

Both methods give 68 cm².

Worked example 4 — area of a T-shape

A T-shape has a horizontal bar (12 cm wide, 4 cm tall) and a vertical stem (4 cm wide, 6 cm tall) dropping from the centre of the bar.

Area of horizontal bar: 12 × 4 = 48 cm²

Area of vertical stem: 4 × 6 = 24 cm²

Total area = 48 + 24 = 72 cm²

Perimeter: The outer edges are (going round clockwise): 12, 4, 4, 6, 4, 6, 4, 4 = 44 cm

Worked example 5 — shape with a rectangular cutout

A rectangle 14 cm × 9 cm has a 3 cm × 5 cm rectangular notch removed from one corner.

Area = (14 × 9) − (3 × 5) = 126 − 15 = 111 cm²

Perimeter: Add all outer edges (the notch creates two new edges of 3 cm and 5 cm):

14 + 9 + 14 + 9 − 3 − 5 + 3 + 5 = outer edges = 14 + 9 + 11 + 4 + 3 + 5 = 46 cm

(Walk the boundary carefully to count every edge exactly once.)

Useful area formulas at KS3

Shape | Formula
Rectangle | length × width
Square | side²
Triangle | ½ × base × height
Parallelogram | base × perpendicular height
Trapezium | ½ × (a + b) × h

Compound shapes in the national curriculum

The DfE's KS3 mathematics programme of study requires pupils to calculate and solve problems involving the perimeters and areas of 2-D shapes, including compound shapes. The AQA GCSE Mathematics specification confirms compound shape problems are a regular component of KS3 and GCSE geometry questions, often requiring pupils to derive missing lengths before calculating.

Common mistakes with compound shapes

Mistake | How it appears | How to avoid it
Including an internal dividing line in the perimeter | Adding an extra edge that is inside the shape | Only count edges on the outer boundary
Missing a side in the perimeter calculation | Perimeter too small | Label all missing sides first and count systematically
Wrong dimensions in the area formula | Using the wrong length for a subdivided rectangle | Draw the split clearly and label each sub-shape's dimensions
Double-counting an edge | Perimeter too large | Walk around the outside systematically, never repeating a side

Frequently asked questions

How do I find the area of a compound shape?

Split the shape into rectangles (and triangles if needed) along natural join lines. Find the area of each piece separately using the relevant formula. Then add all the individual areas together to find the total. Alternatively, find the area of the enclosing rectangle and subtract the area of any cutout pieces — both approaches give the same answer.

How do I find the perimeter of a compound shape?

Add the lengths of all the outer edges. Before you start, label any missing side lengths by using the fact that opposite sides of a rectilinear shape have equal total lengths when extended. Walk around the shape methodically, including every outer edge exactly once, and avoid including any internal dividing lines you drew to split the shape.

What is the difference between area and perimeter?

Perimeter is the total distance around the outside edge of a shape, measured in units (cm, m, etc.). Area is the amount of space enclosed inside the shape, measured in square units (cm², m², etc.). A common error is confusing the two — remember that area involves multiplying two lengths together, so the units are always squared.

Do I always need to split a compound shape into rectangles?

For shapes made entirely of right angles (rectilinear shapes), splitting into rectangles is the standard approach. For shapes that include triangles, semicircles, or other curves, use the appropriate formula for each component. The choice of where to split is flexible — choose the split that gives you the most straightforward calculation.

For Socratic maths tutoring on geometry and shape — visit aitutors.me.