Compound shapes and parts of shapes

Revise Compound shapes and parts of shapes for Mathematics D 4024 (O Level) — revision notes, 4 practice questions and instant AI marking.

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Cambridge O Level Mathematics (Syllabus D) 4024

Compound shapes and parts of shapes

The big idea: A compound shape is two or more simple shapes put together, and a part of a shape is a simple shape with a piece taken away. Split it, or find the whole and subtract.

Summary — What This Chapter Covers

Split a compound 2D shape into rectangles, triangles, circles and sectors; add or subtract their areas.
For a perimeter, go round the outside only: no internal lines.
Segment area = sector area − triangle area.
Compound solids: add the volumes; for surface area, count only the outside surfaces.
Hemisphere: V = ⅔πr³, curved surface area = 2πr².
A frustum is a large cone (or pyramid) minus a small similar one.

1. Compound 2D Shapes

Syllabus 5.5 · Paper 1 & 2

Add the areas; walk round the outside

For the area, split the shape into pieces you have formulas for. For the perimeter, list every outside edge, including curved ones, and leave out the lines where the pieces join.

Worked Example

A window is a rectangle 1.2 m wide and 1.5 m tall with a semicircle on top, whose diameter is the top edge of the rectangle. Find (a) the area of the window, (b) its perimeter. Give each answer correct to 3 significant figures.

2. Parts of Shapes

Syllabus 5.5 · Paper 1 & 2

Whole minus the piece

A shaded region is often easier as "large shape minus small shape". A segment is a sector minus the triangle formed by the two radii and the chord. A ring (annulus) is a large circle minus a small one.

Segment area of segment = (θ ÷ 360) × πr² − ½r² sin θ

Worked Example

A chord AB subtends an angle of 90° at the centre O of a circle of radius 10 cm. Find the area of the minor segment, correct to 3 significant figures.

Practice Question 1

(a) A ring is the region between two circles with the same centre, radii 7 cm and 4 cm. Find its area in terms of π. (b) A square has side 10 cm. A quarter circle of radius 10 cm is drawn with its centre at one corner of the square. Find the area of the square outside the quarter circle, correct to 3 significant figures.

Practice Question 2

Find the area of the minor segment cut off by a chord that subtends 60° at the centre of a circle of radius 10 cm. Give your answer correct to 3 significant figures.

3. Hemispheres and Compound Solids

Syllabus 5.5 · Paper 1 & 2

Add volumes, count outside surfaces

A hemisphere is half a sphere: half the volume and half the curved surface. A solid hemisphere also has a flat circular face. When solids are joined, the faces where they touch are hidden and are not part of the surface area.

Hemisphere (from the sphere formulas) V = ⅔πr³ curved surface = 2πr² solid hemisphere total = 3πr²

Worked Example

A solid is a cylinder of radius 3 cm and height 10 cm with a hemisphere of radius 3 cm on top. Find (a) its volume, (b) its total surface area. Give each answer in terms of π.

Practice Question 3

A ball of radius 3 cm is dropped into a cylinder of radius 6 cm containing water, and is completely covered. By how much does the water level rise?

4. Frustums

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Big cone minus small cone
Hollow and cut solids

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Compound shapes and parts of shapes