Key Stage 4 maths · Geometry and measures

GCSE circles, sectors, 3D solids and bearings: a parent’s guide

At GCSE, circles stop being just a shape and become a toolkit. Children name the parts of a circle precisely, find arc lengths and the areas of sectors, calculate surface areas and volumes of spheres, pyramids, cones and composite solids, draw plans and elevations of 3D shapes, and use three-figure bearings to describe direction. This builds on the circle, area and volume work from earlier years and is taught to all pupils in Years 10 and 11.

Circles, solids and bearings, explained

Circles come with a set of words that GCSE questions expect children to use precisely. The centre is the middle point, the radius runs from the centre to the edge, and the diameter passes through the centre and is twice the radius. The circumference is the distance around the edge, a chord is a straight line joining two points on the edge, an arc is a part of the circumference, and a tangent is a straight line that touches the circle at exactly one point. A segment is the region between a chord and an arc. The circumference is found with πd (or 2πr) and the area with πr². Questions often ask for an answer “in terms of π”, which means leaving π in the answer, such as 12π cm², instead of using a calculator value.

A sector is a slice of a circle, like a slice of pizza, bounded by two radii and an arc. The method is always the same: work out what fraction of the whole circle the sector is, by writing the angle over 360, then take that fraction of the circumference to get the arc length, or of πr² to get the area. Children also work backwards, for example finding the angle of a sector when its area and radius are given, or finding the radius from an arc length.

Three-dimensional work extends the volume and surface area they know from prisms. A pyramid has a volume of one third of its base area multiplied by its perpendicular height, and a cone holds one third of the cylinder that fits around it. The volume of a sphere is 4/3 πr³ and its surface area is 4πr². The curved surface of a cone has area πrl, where l is the slant height, the distance from the tip down the side to the edge of the base. Composite solids are made by joining shapes, such as a hemisphere on top of a cylinder, so children split them into parts, calculate each, and think carefully about which faces are really on the outside. Which formulae must be memorised and which are provided in the exam is something your child’s teacher will explain.

Plans and elevations are the views of a 3D shape from above (the plan), from the front (the front elevation) and from the side (the side elevation). Children draw these on squared paper, sketch solids on isometric paper, and work backwards from a set of views to the solid. Bearings describe direction as an angle measured clockwise from north, always written with three figures, so east is 090° and south-west is 225°. The bearing of B from A means standing at A. Bearings problems draw on parallel line angle facts, scale drawing, Pythagoras’ Theorem and trigonometry, so they are a good place to see earlier topics being combined.

Words your child will hear

Arc: A part of the circumference of a circle, between two points on the edge.
Chord: A straight line joining two points on a circle. The longest chord is a diameter.
Sector: The slice of a circle between two radii and the arc that joins them.
Segment: The region of a circle cut off by a chord, between the chord and its arc.
Tangent: A straight line that touches a circle at one point without crossing into it.
Bearing: A direction given as an angle clockwise from north, written with three figures, such as 072°.
Elevation: A view of a 3D shape from the front or the side, drawn as a flat shape.

What the National Curriculum says

In Key Stage 4, pupils should be taught to:

Identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment
Calculate arc lengths, angles and areas of sectors of circles
Calculate surface areas and volumes of spheres, pyramids, cones and composite solids
Construct and interpret plans and elevations of 3D shapes
Interpret and use bearings

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

A sector of a circle has radius 6 cm and angle 120°. Find its arc length and its area, first in terms of π and then to 2 decimal places.

Find the fraction of the circle. The angle is 120° out of 360°, so the sector is 120/360 = 1/3 of the whole circle.
Arc length: the whole circumference is 2 × π × 6 = 12π cm. One third of that is 4π cm.
```
arc = 1/3 × 2 × π × 6
    = 1/3 × 12π
    = 4π cm
    = 12.57 cm (2 d.p.)
```

Area: the whole circle has area π × 6² = 36π cm². One third of that is 12π cm².

area = 1/3 × π × 6²
     = 1/3 × 36π
     = 12π cm²
     = 37.70 cm² (2 d.p.)

Check that it is sensible. The full circle has an area of about 113 cm², and a third of that is about 38 cm². The units are cm for the arc and cm² for the area.

Arc length = 4π cm ≈ 12.57 cm. Area = 12π cm² ≈ 37.70 cm².

Common mistakes and how to help

Measuring a bearing anticlockwise, or from the wrong place

Earlier angle work is often measured anticlockwise from a horizontal line, so the habit carries over. “The bearing of B from A” also gets reversed, with the angle measured at B instead of at A.

What helps: Always draw a north line at the point you are starting from, then measure clockwise. Write the answer with three figures, so 72° becomes 072°.

Finding a back bearing by subtracting from 360°

If the bearing of B from A is 070°, children often write 290° (360 − 70) for the bearing of A from B. The correct answer is 250°, because the two north lines are parallel and the direction is exactly reversed.

What helps: Draw both north lines and use the parallel line angle facts. As a rule, add 180° if the bearing is less than 180°, and subtract 180° if it is more. A quick sketch shows whether the answer points the opposite way.

Using the diameter as the radius, or mixing up area and circumference

A circle with diameter 10 cm has area π × 5² ≈ 78.5 cm², but many children calculate π × 10² ≈ 314 cm². The two formulae both involve π, so they are easy to swap.

What helps: Halve the diameter before you do anything else, and write r = on the diagram. Check units: lengths are in cm, areas in cm². Circumference measures a distance and area measures a surface.

Using the slant height in a volume formula, or leaving out the one third

A cone with radius 3 cm and perpendicular height 4 cm has a slant height of 5 cm. Using 5 instead of 4 gives 15π cm³ instead of the correct 12π cm³. The one third is easily forgotten because it is missing from prism and cylinder formulae.

What helps: Sketch the right-angled triangle inside the cone to separate the perpendicular height from the slant height. Remember that a cone or pyramid is one third of the cylinder or prism with the same base and height.

How to help at home

Measure round things. Wrap string around a tin, a plate and a bin lid, then measure the string and the width. Dividing circumference by diameter gives just over 3 every time. Ask your child to predict the circumference of a larger round object before measuring it.
Walk a bearing. With a simple compass in a garden or park, well away from roads, pick two landmarks. Ask for the bearing of one from the other, then the back bearing. Check it by facing the second landmark and reading the compass.
Cut it into sectors. Mark a paper plate into equal slices with a ruler, or use a round cake or pizza at teatime. Eight slices have an angle of 45° each. Ask what fraction of the whole a slice of 90° or 120° would be, and what that means for the crust length. An adult should do any cutting.
Draw plans and elevations. Build a small model from blocks or boxes. Sketch it from above, from the front and from the side, then swap sketches with someone who rebuilds the model from your drawings. The mistakes that come up are exactly the ones exam questions test.
Spot cones and pyramids. Ice cream cones, traffic cones, some tents and some roofs are everyday cones and pyramids. Estimate the volume of the box or cylinder that would fit around each one, then take a third. It builds a feel for whether a calculated answer is reasonable.

Builds on

Key Stage 3: Perimeter, area and volume
Key Stage 3: Constructions and scale drawings
Key Stage 3: Transformations and enlargement
Key Stage 3: Angles, parallel lines and polygons
Key Stage 3: Pythagoras and trigonometry