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17.7 Spheres and Composite Solids

GCSE Maths Foundation · Edexcel · Chapter 17: Perimeter, Area and Volume 2

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: volume and surface area of spheres and hemispheres, and solids made from several shapes. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Work out the volume and surface area of a sphere using given formulae
  • Work out the volume and surface area of a hemisphere
  • Find the volume of a composite solid by adding the volumes of its parts
  • Find the surface area of a composite solid, leaving out faces that are joined together

Key words

  • Sphere: A perfectly round ball. Every point on its surface is the same distance (r) from the centre.
  • Hemisphere: Half a sphere, with a flat circular face.
  • Composite solid: A solid made by joining simpler solids, such as a cone on a cylinder.
  • Curved surface: The part of the surface that is not flat.

Spheres

In the exam the sphere formulae are given to you. You need to choose the right one and substitute carefully: cube the radius for volume, square it for surface area.

Formulae: Volume of a sphere = 4/3 × π × r³ Surface area of a sphere = 4 × π × r²

Worked example: a ball with radius 6 cm

V = 4/3 × π × 6³ = 4/3 × π × 216 = 288π = 904.778… cm³
SA = 4 × π × 6² = 144π = 452.389… cm²

Answer: Volume 904.8 cm³, surface area 452.4 cm²

On a calculator, type 4 ÷ 3 × π × 6³ in one go, or work out 6³ first. Do not round 4/3 to 1.3.

Hemispheres

A hemisphere has half the volume of a sphere. Its surface is half the sphere's curved surface plus a flat circle.

  • Volume; Formula ½ × 4/3 × π × r³ = 2/3 × π × r³; r = 4 cm 2/3 × π × 64 = 134.0 cm³
  • Curved surface; Formula ½ × 4 × π × r² = 2 × π × r²; r = 4 cm 32π = 100.5 cm²
  • Total surface (solid); Formula 2πr² + πr² = 3 × π × r²; r = 4 cm 48π = 150.8 cm²

Worked example: a solid hemisphere, radius 4 cm

Curved part: 2 × π × 4² = 32π. Flat circle: π × 4² = 16π.
Total surface area = 32π + 16π = 48π = 150.796… cm²

Answer: 150.8 cm²

Volume of composite solids

Split the solid into parts you know, find each volume, then add. Write each part as a multiple of π first: it keeps the numbers exact and easy to add.

Worked example: cylinder + hemisphere

Cylinder: π × 3² × 10 = 90π. Hemisphere: 2/3 × π × 3³ = 18π.
Total = 90π + 18π = 108π = 339.292… cm³

Answer: 339.3 cm³

Worked example: cone on a cylinder

A pencil cap is a cylinder (radius 2 cm, height 9 cm) with a cone of height 4 cm on top.
Cylinder: π × 2² × 9 = 36π. Cone: ⅓ × π × 2² × 4 = 16π/3 = 5⅓π.
Total = 41⅓π = 129.852… cm³

Answer: 129.9 cm³

Surface area of composite solids

Add the areas of the surfaces you can touch on the outside. Where two parts are joined, that circle is hidden inside, so leave it out.

Worked example: cylinder + hemisphere (the solid (drawn in the printable lesson notes))

Base circle: π × 3² = 9π. Curved side of cylinder: 2 × π × 3 × 10 = 60π.
Curved hemisphere: 2 × π × 3² = 18π. (No circle where they join.)
Total = 9π + 60π + 18π = 87π = 273.318… cm²

Answer: 273.3 cm²

Common mistakes

  • Squaring the radius in the sphere volume. Volume needs r³.
  • Using the diameter in the formulae. Halve it first.
  • Forgetting the flat circle on a solid hemisphere, or including a joined face in a composite solid.
  • Rounding each part before adding. Keep answers in terms of π (or unrounded) until the end.
  • Rounding 4/3 to 1.3, which gives a wrong answer.

Revision questions

Try these, then check your answers below each one.

Question 1. A sphere has radius 3 cm. Work out its volume. Give your answer to 1 decimal place. (Volume of a sphere = 4/3π r³)

Answer: 113.1 cm³

Working: 4/3 × π × 27 = 36π = 113.097… = 113.1 cm³.

Question 2. A sphere has radius 7 cm. Work out its surface area. Give your answer to 1 decimal place. (Surface area = 4π r²)

Answer: 615.8 cm²

Working: 4 × π × 49 = 196π = 615.752… = 615.8 cm².

Question 3. What is the volume of a hemisphere with radius 6 cm, in terms of π? (Choose from: 144π cm³, 288π cm³, 72π cm³, 216π cm³.)

Answer: 144π cm³

Working: 2/3 × π × 216 = 144π cm³.

Related lessons

Continue in the full course

This lesson is section 17.7 (EGCMF1707) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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