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Volume of a frustum

Free Printable Volume of a frustum PDF Worksheets, Revision Guide & GCSE Past Paper Questions (Grade 8+)

Download free volume of a frustum worksheets (PDF), GCSE maths past paper questions and worked solutions for Volume of a frustum (8P). This is a Grade 8+ geometry topic from the GCSE questions by topic collection for Foundation and Higher tier GCSE Maths (Edexcel, AQA, OCR and IGCSE).

The free PDF worksheets table below brings together printable question booklets (Qs) and worked answers or mark schemes (Ans) from Corbett Maths, Maths Made Easy, Dr Austin, M4ths.com, E Classroom and PMT, along with past paper question booklets.

Below the worksheet links you can read our volume of a frustum revision notes and worked examples with diagrams, try real GCSE past paper volume of a frustum questions with step-by-step solutions, or download a PDF of all volume of a frustum worksheet links.

Free Volume of a frustum PDF Worksheets & Answers

Grade 8+ • Topic 8P
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Grade 8+ topics
8A - Quad. sim. equations8B - Transforming graphs y=f(x)
8C - Algebraic proof8D - Completing the square
8E - Quadratic inequalities8F - Quadratic sequences
8G - Velocity time graphs8H - Circles and tangents
8J - Circle theorem proof8K - Vector proof
8L - Probability equation questions8M - Invariant points
8N - Spheres, cones and cylinders8P - Volume of a frustum
GCSE Maths Grade 8+ Revision Notes & Method

Guide to Volume of a frustum: Explanation, Diagrams & Worked Examples

Download Printable One Page Guide (PDF)

Explanation:

A frustum is the solid remaining when a smaller cone (or pyramid) is sliced off the top of a larger cone parallel to its base. Because the removed top cone and the original full cone are mathematically similar, the ratio of their heights equals the ratio of their radii: hH = rR.

Use similar triangles first if the height of the removed top cone is not given, then calculate: Volume of frustum = Volume of large cone − Volume of small cone = 13πR2H − 13πr2h.

Example 1: A cone of radius 9 cm and height 15 cm has a smaller cone of radius 3 cm and height 5 cm removed from its top. Find the exact volume of the frustum.

h = 5 r = 3 9 cm 15 cm

Solution:

Vlarge = 13π(9)2(15) = 405π;  Vsmall = 13π(3)2(5) = 15π
Vfrustum = 405π − 15π = 390π cm3

Example 2: A frustum has top radius 4 cm, base radius 10 cm, and vertical height 9 cm. Find the volume of the frustum (3 s.f.).

h 9 4 10 cm

Solution:

hh + 9 = 410 ⇒ 10h = 4h + 36 ⇒ h = 6 cm (H = 15 cm)
V = 13π(102×15 − 42×6) = 468π = 1470 cm3 (3 s.f.)
Scanned GCSE Exam Questions & Worked Solutions

GCSE Past Paper Volume of a frustum Questions & Answers

Past Paper Example 1 IGCSE Nov 2023 2H Q23 5 marks
Full Paper (PDF) • Mark Scheme (PDF)
Here is a frustum of a cone.
The frustum is made by removing a small cone from a similar large cone.
x cm6 cm15 cmDiagram NOTaccurately drawn
The height of the large cone is 15 cm.
The radius of the base of the large cone is 6 cm.
The radius of the base of the small cone is x cm.
Given that the volume of the frustum is 421225π cm3
work out the value of x
Show clear algebraic working.
Worked Solution & Mark Scheme (5 marks)
Volume of large cone = (1/3)π(6²)(15) = 180π cm³ M2
Volume of small cone = 180π − (4212/25)π = (288/25)π = 11.52π cm³ M2
By similarity, (x/6)³ = 11.52π / 180π = 288 / 4500 = 8 / 125 ⇒ x/6 = 2/5 ⇒ x = 2.4. A1
Answer: x = 2.4
Past Paper Example 2 AQA Nov 2022 3H Q23 5 marks
Full Paper (PDF) • Mark Scheme (PDF)
Alec makes a bowl for dog food from a solid wooden cone.
The sketches show how the bowl is made.
The cone has radius 9 cm and perpendicular height 30 cm
A smaller cone, with radius 6 cm, is removed.
30 cm9 cm6 cmNot drawnaccurately
@@TABULAR0@@
A hemisphere with radius 6 cm is then removed.
@@TABULAR1@@
Work out the volume of the remaining wood that forms the bowl.
Worked Solution & Mark Scheme (5 marks)
By similar cones: height of removed cone = 30 × (6/9) = 20 cm M1
Volume of large cone = 1/3 × π × 9² × 30 = 810π cm³ M1
Volume of small cone = 1/3 × π × 6² × 20 = 240π cm³ M1
Volume of hemisphere = 2/3 × π × 6³ = 144π cm³ M1
Remaining volume = (810 – 240 – 144)π = 426π = 1338.3 cm³. A1
Answer: Answer 1338 cm³
Past Paper Example 3 GCSE Nov 2023 3H Q19 4 marks
Full Paper (PDF) • Mark Scheme (PDF)
24 cm5 cmVolume of sphere = ⁴⁄₃ πr³rVolume of cone = ⅓ πr²hlhrThe diagram above shows a frustum F of a cone.The frustum is made by removing a cone with height 10 cm from a solid cone withheight 15 cm and base diameter 24 cm.The solid S is made by removing F from a solid hemisphere as shown in thediagram below.24 cm5 cm
Work out
A frustum F is made by removing a cone of height 10 cm from a cone of height 15 cm and base diameter 24 cm. Solid S is made by removing F from a hemisphere of diameter 24 cm. Calculate the volume of solid S to 3 s.f.
The hemisphere has diameter 24 cm.
Calculate the volume of solid **S**.
Give your answer correct to 3 significant figures.
Worked Solution & Mark Scheme (4 marks)
Small cone radius = (10/15) × 12 = 8 cm M1
Volume of F = (1/3)π(12² × 15 – 8² × 10) = (1/3)π(2160 – 640) = (1520/3)π ≈ 1591.74 cm³ M1
Hemisphere volume = (2/3)π(12³) = 1152π ≈ 3619.11 cm³ M1
Volume of S = 3619.11 – 1591.74 = 2027.37 cm³ ≈ 2030 cm³. A1
Answer: 2030 cm³

← 8N Spheres, cones and cylinders

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