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17.6 Pyramids and Cones

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 pyramids and cones. 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 of a pyramid using ⅓ × base area × height
  • Work out the surface area of a pyramid from its faces
  • Work out the volume and surface area of a cone using given formulae
  • Use Pythagoras to find a slant height, and work backwards to find a height

Key words

  • Pyramid: A solid with a flat base and triangular faces that meet at a point called the apex.
  • Apex: The top point of a pyramid or cone.
  • Perpendicular height: The vertical height from the base straight up to the apex, at right angles to the base.
  • Slant height (l): The distance down the sloping side, from the apex to the edge of the base.
  • Cone: A solid with a circular base and a curved surface rising to an apex.

Volume of a pyramid

A pyramid fits inside a prism with the same base and height. It takes exactly three pyramids of water to fill the prism, so the pyramid has one third of the volume.

Formula: Volume of a pyramid = ⅓ × area of base × perpendicular height

Worked example: square-based pyramid

Base area = 6 × 6 = 36 cm². V = ⅓ × 36 × 9 = 108 cm³

Answer: 108 cm³

Worked example: rectangular base

A pyramid has a rectangular base 5 cm by 8 cm and height 12 cm.
Base area = 40 cm². V = ⅓ × 40 × 12 = 160 cm³

Answer: 160 cm³

Surface area of a pyramid

Add the area of the base and the area of every triangular face. For a triangular face you need the slant height of that face, not the vertical height of the pyramid.

Worked example: square-based pyramid

A pyramid has a square base of side 6 cm. Each triangular face has height 7 cm (slant height).
Base: 6 × 6 = 36 cm². One triangle: ½ × 6 × 7 = 21 cm². Four triangles: 4 × 21 = 84 cm².
Total = 36 + 84 = 120 cm²

Answer: 120 cm²

Volume of a cone

A cone is like a pyramid with a circular base, so its volume is one third of a cylinder with the same radius and height. In the exam the cone formulae are given to you when you need them.

Formula: Volume of a cone = ⅓ × π × r² × h

Worked example: cone volume

V = ⅓ × π × 5² × 12 = ⅓ × 300π = 100π = 314.159… cm³

Answer: 314.2 cm³ (or exactly 100π cm³)

Surface area of a cone

Formulae: Curved surface area = π × r × l (l = slant height) Total surface area = π × r × l + π × r² (curved part + circular base)

If you know r and h but not l, use Pythagoras: the radius, height and slant height make a right-angled triangle, so l² = r² + h².

Worked example: total surface area

r = 5 cm, h = 12 cm. l² = 5² + 12² = 25 + 144 = 169, so l = √169 = 13 cm.
Curved: π × 5 × 13 = 65π. Base: π × 5² = 25π. Total = 90π = 282.743… cm²

Answer: 282.7 cm² (or 90π cm²)

Working backwards

If you know the volume, rearrange: height = 3 × volume ÷ base area.

Worked example: missing height

A pyramid has volume 200 cm³ and a rectangular base 10 cm by 6 cm. Find its height.
Base area = 60 cm². 200 = ⅓ × 60 × h = 20h, so h = 200 ÷ 20 = 10

Answer: 10 cm

Common mistakes

  • Forgetting the ⅓ in the volume of a pyramid or cone.
  • Using the slant height instead of the perpendicular height for volume.
  • Leaving out the base when asked for the total surface area.
  • Using the diameter instead of the radius in the cone formulae.
  • Counting the wrong number of triangular faces (a square-based pyramid has 4).

Revision questions

Try these, then check your answers below each one.

Question 1. A square-based pyramid has base side 5 cm and perpendicular height 12 cm. Work out its volume.

Answer: 100 cm³

Working: 1/3 × 25 × 12 = 100 cm³.

Question 2. A cone has radius 6 cm and height 10 cm. Work out its volume. Give your answer to 1 decimal place. (Volume of a cone = 1/3π r² h)

Answer: 377.0 cm³

Working: 1/3 × π × 36 × 10 = 120π = 376.99… = 377.0 cm³.

Question 3. Which formula gives the volume of a pyramid? (Choose from: 1/3 × base area × height, base area × height, 1/2 × base area × height, 3 × base area × height.)

Answer: 1/3 × base area × height

Working: V = 1/3 × base area × height.

Related lessons

Continue in the full course

This lesson is section 17.6 (EGCMF1706) 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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