Area of Compound Shapes
x

Revision notes for Edexcel GCSE Maths Area of Compound Shapes. Open the guide for explanations and worked examples. Written against the Edexcel GCSE Maths (1MA1) specification, so the content matches what's examinable rather than general Maths background.

Area of Compound Shapes

What you'll learn

  • Recall the area rules for rectangles and triangles.
  • Break compound shapes into simpler shapes.
  • Find missing side lengths in right-angled shapes.
  • Subtract holes or unshaded parts, and use area in money questions.

1. Area: the space inside a shape

Definition

Area

Area is the amount of space inside a 2D shape. It is measured in square units, such as cm² or m².

A formula is a rule you can use for a calculation. For a rectangle with length lll and width www, use A=l×wA = l \times wA=l×w.

A right angle is an angle of 90°. Rectangles have four right angles.

Example

Area of a rectangle

  1. A rectangle is 9 cm long and 4 cm wide.

Rectangle with length 9 cm and width 4 cm, showing the two dimensions used for area.

  1. Use the rectangle formula: A=l×wA = l \times wA=l×w.

  2. Substitute the numbers: A=9×4=36A = 9 \times 4 = 36A=9×4=36.

  3. The area is 36 cm².

Common Mistake

Measuring the diagram

If a diagram says it is not accurately drawn, do not measure it with a ruler. Use the numbers written on the diagram.

2. Triangles: half a rectangle

For a triangle, you need the base and the height.

Definition

Perpendicular height

The perpendicular height is the straight height that meets the base at a right angle. It is not always the sloping side.

Triangle showing that the perpendicular height meets the base at 90°, while the sloping side is not the height.

For a triangle with base bbb and perpendicular height hhh, use A=12×b×hA = \frac{1}{2} \times b \times hA=21​×b×h.

Example

Area of a triangle

  1. A triangle has base 12 cm and perpendicular height 7 cm.

Triangle with base 12 cm and perpendicular height 7 cm for applying the triangle area formula.

  1. Use the triangle formula: A=12×b×hA = \frac{1}{2} \times b \times hA=21​×b×h.

  2. Substitute the values: A=12×12×7=42A = \frac{1}{2} \times 12 \times 7 = 42A=21​×12×7=42.

  3. The area is 42 cm².

Common Mistake

Using the sloping side

For triangle area, only use the height that is at right angles to the base. The slanted side is not usually the height.

3. What is a compound shape?

Definition

Compound shape

A compound shape is a shape made from two or more simpler shapes, such as rectangles, triangles or trapeziums.

The main skill is to draw a sensible line to split the shape into parts you already know how to find.

Key Idea

Split, find, add

If the compound shape is made by joining shapes together, find each area separately and add them.

Example

Rectangle joined to a right-angled triangle

  1. A 6-sided shape is made from a rectangle and a right-angled triangle. The rectangle is 2 cm by 11 cm. The triangle has base 6 cm and height 7 cm.

Compound shape split into a 2 cm by 11 cm rectangle and a right-angled triangle with base 6 cm and height 7 cm.

  1. Find the rectangle area: A=2×11=22A = 2 \times 11 = 22A=2×11=22.

  2. Find the triangle area: A=12×6×7=21A = \frac{1}{2} \times 6 \times 7 = 21A=21​×6×7=21.

  3. Add the two areas: 22+21=4322 + 21 = 4322+21=43.

  4. The total area is 43 cm².

4. Finding missing lengths

A rectilinear shape is made from horizontal and vertical sides, so its corners are right angles. These shapes often have missing lengths that you can work out by comparing opposite sides.

Tip

Opposite totals match

In a closed right-angled shape, the total distance across the top matches the total distance across the bottom. The total distance up matches the total distance down.

Example

An L-shape made of rectangles

  1. An L-shape has bottom length 14 cm, left height 9 cm, top length 6 cm, and a vertical drop of 4 cm near the top right.

L-shape with given outer lengths and the missing width and height that can be found from opposite totals.

  1. Find the missing width: 14−6=814 - 6 = 814−6=8.

  2. Find the lower height: 9−4=59 - 4 = 59−4=5.

  3. Split the shape into two rectangles: one is 6 cm by 9 cm, and the other is 8 cm by 5 cm.

  4. Find their areas: 6×9=546 \times 9 = 546×9=54 and 8×5=408 \times 5 = 408×5=40.

  5. Add them: 54+40=9454 + 40 = 9454+40=94.

  6. The area is 94 cm².

Common Mistake

Multiplying every label

Do not multiply all the numbers you see. Choose a rectangle or triangle, use only the lengths needed for that part, then move to the next part.

5. Shaded areas and holes: subtract

A shaded region is the grey part you are being asked to find. If a shape has a hole or an unshaded part inside it, find the big area first, then subtract the unwanted area.

For shaded area, think Ashaded=Alarge−AremovedA_{\text{shaded}} = A_{\text{large}} - A_{\text{removed}}Ashaded​=Alarge​−Aremoved​.

Example

Shaded area outside a rectangle inside a triangle

  1. A triangle has base 14 cm and height 8 cm. Inside it is an unshaded rectangle measuring 5 cm by 3 cm.

Shaded triangular region with an unshaded 5 cm by 3 cm rectangle removed from inside.

  1. Find the triangle area: A=12×14×8=56A = \frac{1}{2} \times 14 \times 8 = 56A=21​×14×8=56.

  2. Find the rectangle area: A=5×3=15A = 5 \times 3 = 15A=5×3=15.

  3. Subtract the rectangle from the triangle: 56−15=4156 - 15 = 4156−15=41.

  4. The shaded area is 41 cm².

6. Right-angled trapeziums

A trapezium is a 4-sided shape with one pair of parallel sides. Parallel sides stay the same distance apart and never meet.

A right-angled trapezium can often be split into a rectangle and a right-angled triangle.

Example

Splitting a trapezium

  1. A right-angled trapezium has top length 8 cm, bottom length 12 cm, and height 5 cm.

Right-angled trapezium split into an 8 cm by 5 cm rectangle and a right-angled triangle with extra bottom width.

  1. Find the extra width on the bottom: 12−8=412 - 8 = 412−8=4.

  2. Split it into a rectangle and a triangle.

  3. Rectangle area: 8×5=408 \times 5 = 408×5=40.

  4. Triangle area: A=12×4×5=10A = \frac{1}{2} \times 4 \times 5 = 10A=21​×4×5=10.

  5. Total area: 40+10=5040 + 10 = 5040+10=50.

  6. The area of the trapezium is 50 cm².

7. Area used with money

Sometimes an area question asks for a cost. Per square metre means “for each 1 m²”.

Always find the area first. Then multiply by the cost per m².

Example

Selling a field

  1. An L-shaped field has bottom length 120 m, left height 80 m, top length 50 m, and short right height 30 m. It is sold for £3 per square metre.

L-shaped field split into two rectangles before multiplying the total area by £3 per square metre.

  1. Find the missing width: 120−50=70120 - 50 = 70120−50=70.

  2. Split the field into two rectangles. Their areas are 50×80=400050 \times 80 = 400050×80=4000 and 70×30=210070 \times 30 = 210070×30=2100.

  3. Add the areas: 4000+2100=61004000 + 2100 = 61004000+2100=6100.

  4. Multiply by the price: 6100×3=183006100 \times 3 = 183006100×3=18300.

  5. The farmer should get £18,300.

Exam technique

In the exam

  1. Draw splitting lines lightly, then label any missing lengths before calculating.

  2. Add areas when shapes are joined; subtract areas when there is a hole or an unshaded part.

  3. Write the correct square units, and for money questions multiply by the price only after finding the area.

Self review

Check yourself

  • Can you explain why a triangle area formula includes 12\frac{1}{2}21​?
  • In an L-shape, how can you find a missing width or height?
  • If a small rectangle is cut out of a large rectangle, should you add or subtract its area?

Recap questions

Test yourself with 5 quick questions on this guide. Answer them all correctly to complete it.

You've reached the end

Test yourself on this topic, or move on to the next guide.

FlashcardsSelf-test with active recall
Frequency TreesUp next

How was this guide?

Area of Compound Shapes Revision Guide

  1. GCSE
  2. /Maths
  3. /Area of Compound Shapes