Solids and circle vocabulary
Name faces, edges, cross-sections and the parts of a circle before calculating with them.
Core and Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Recognise triangles, quadrilaterals and perpendicular lines.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
How many faces does a cube have?
What you will learn
- Distinguish common solids and their surfaces.
- Identify chords, tangents, arcs, sectors and segments.
- Explain why a diameter is a special chord.
Describe a solid
A cube has six square faces; a cuboid has six rectangular faces. A prism has a uniform cross-section along its length: a triangular prism has two matching triangular ends and three rectangular side faces when it is a right prism. A pyramid joins a polygonal base to one apex with triangular faces. Faces are flat surfaces, edges join surfaces, and vertices are corners.
A cylinder has two parallel circular ends and a curved surface. A cone has one circular base, a curved surface and an apex. A sphere has one curved surface; a hemisphere is half a sphere with a circular flat base. A frustum is the part of a cone or pyramid left after cutting off its top parallel to the base. State whether you mean a flat face or a curved surface when describing a solid.
Separate the parts of a circle
The radius joins the centre to the circumference. A diameter passes through the centre between two circumference points and has length twice the radius. A chord joins any two circumference points; a diameter is the longest chord. A tangent touches the circle at one point, whereas a secant crosses it at two points.
An arc is part of the circumference. Two endpoints usually define a shorter minor arc and a longer major arc. A sector is bounded by two radii and an arc, like a slice; a segment is bounded by a chord and an arc. A semicircle is half the circle cut by a diameter. Identify the boundary before deciding whether a shaded region is a sector or a segment.
Pause and explain
Which boundary describes a sector?
Worked example
A circle has radius 7 cm. A shaded region is bounded by a chord and its minor arc. Name the region and give the diameter.
Show the worked solution
- A chord with an arc bounds a segment, rather than a sector.
- The shorter arc makes this the minor segment.
- The diameter is twice the radius: 2 × 7 = 14 cm.
Answer Minor segment; diameter 14 cm.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
List the faces of a right triangular prism and explain why it is not a triangular pyramid.
Give me a hint
Compare its ends with a pyramid's apex.
Compare my reasoning
- There are two matching triangular ends.
- Three rectangular faces join the corresponding sides.
- A pyramid has one base and an apex; this solid keeps the same triangular cross-section.
Two triangles and three rectangles; a uniform cross-section, not one apex.
Look for these in your work
- Counted both end faces.
- Used the solid's defining property.
A chord goes through a circle's centre. The circle has diameter 18 cm. Name the chord and find the radius.
Give me a hint
The centre condition distinguishes this chord.
Compare my reasoning
- A chord through the centre is a diameter.
- A radius is half the diameter.
- 18 ÷ 2 = 9 cm; include the length unit.
Diameter; radius 9 cm.
Look for these in your work
- Applied the centre condition.
- Halved rather than doubled the given diameter.
A lampshade is made by cutting the top off a cone parallel to its base. Name the remaining solid and describe its two ends.
Give me a hint
The cut creates a smaller parallel circular end.
Compare my reasoning
- The remaining solid is a conical frustum.
- Its large and small ends are parallel circles.
- A curved surface joins them; the cone's original apex has been removed.
A conical frustum with two parallel circular ends of different sizes.
Look for these in your work
- Used the term frustum accurately.
- Distinguished the remaining solid from a complete cone.
Common mistakes
- Counting edges instead of faces.
- Confusing a sector with a segment.
- Calling any line near a circle a tangent without the one-point contact condition.
What can you explain now?
Does every chord pass through the centre?
Compare with the explanation
No; only a diameter does.
A chord requires two endpoints on the circumference, without requiring the centre to lie on it.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Sketch and label a chord, a diameter, a tangent, a sector and a segment tomorrow. Describe one prism and one pyramid aloud.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.