IGCSE Maths / Circles, cylinders, cones and spheres

Compound area with circles revision notes

Edexcel International GCSE Maths (4MA1), chapter 12: Circles, cylinders, cones and spheres, specification point 4.9e-f. Free lesson.

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Compound areas are found by splitting a shape into familiar parts, finding each area, then adding joined pieces or subtracting holes. Getting the radius right and halving the area of a semicircle are key to avoiding large errors.

Split a compound shape into familiar pieces

A compound shape can be worked out by separating it into shapes whose areas you already know. Find each area on its own, then combine them according to how the parts fit together. A rectangle with a semicircle joined to it contains a rectangle and half a circle. A shape with a circular hole contains a larger shape with a circle removed.

Find the radius before using the circle formula

The circle formula uses the radius, not the diameter. If a semicircle sits on a side, that side is its diameter, so halve it to find the radius. Using the diameter as the radius makes the circle area four times too large because the radius is squared. For a diameter d, the radius is r = d ÷ 2. The area of a semicircle is 1/2 × π × r².

Add the area of a semicircle to a rectangle

Work out the rectangle and semicircle separately. The semicircle contributes only half the area of a full circle, so show the 1/2 in your working before calculating. Add the two areas because the semicircle is joined on.

Subtract a circular hole

For a shape with a hole, first find the area of the whole outside shape. Then find the area of the removed circle and subtract it. This time use the full circle formula, because the hole is a whole circle.

Choose addition or subtraction from the diagram

Read the shape carefully to decide whether each piece is joined on or cut out. A joined piece increases the area, while a hole reduces it. The method marks come from showing the correct areas and whether you add or subtract them. A rectangle with a semicircle on top has area lw + 1/2πr², not lw + πr². A semicircle uses half a circle's area, but a circular hole uses the full circle's area.