Year 9 · Unit 4 Cylinders, cones and spheres · Lesson 3 of 6

How many pyramids fill the prism around them?

Pour pyramid-fuls of sand into a prism with the same base and height, and count how many it takes. The count gives the volume formula, and Pythagoras gives the slant height you need for the sloping faces.

Geometry and TrigonometryNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 7 (volume of prisms): a triangular prism has a cross-section of area 12 cm² and is 10 cm long. What is its volume?
    Answer120 cm³. Area of the end times length: 12 × 10 = 120.
  2. From Year 9 (Pythagoras): a right-angled triangle has legs 6 cm and 8 cm. What is the hypotenuse?
    Answer10 cm. 6² + 8² = 36 + 64 = 100, and the square root of 100 is 10.
  3. Work out one third of 7 × 9.
    Answer21. 7 × 9 = 63, and 63 ÷ 3 = 21.
Words used in this lesson
pyramid
a solid with a flat base and sloping faces that meet at a point. For example: a square pyramid.
apex
the top point where the sloping faces meet. For example: the tip of a pyramid.
perpendicular height
the height straight up from the base to the apex. For example: 12 m.
slant height
the height of a sloping face, measured up the middle of the face. For example: 13 m.
face
one flat surface of a solid. For example: a triangle on a pyramid.
Pythagoras' theorem
in a right-angled triangle, the squares of the two short sides add to the square of the longest. For example: 12² + 5² = 13².

A box of sand and a pyramid of sand

A museum in Cairo has a clear plastic box with a square base, and a hollow pyramid of the same plastic with exactly the same base and the same height. A guide fills the pyramid with sand and tips it into the box, again and again, until the box is full.

Predict first: how many pyramids of sand fill the box? Two, three, four? Write your guess and a reason. Then use the first figure, and the second figure for the part that is not obvious: the slant height of the sloping faces.

6 cm
6 cm
0
216 cm³volume of the prism
pour to find outvolume of the pyramid
0 cm³poured so far

Base side 6 cm and height 6 cm, nothing poured yet: the prism holds 6 × 6 × 6 = 216 cm³.

Show me the steps
  1. Base side 6 cm and height 6 cm, nothing poured yet: the prism holds 6 × 6 × 6 = 216 cm³. Pour in one pyramid-ful and see how much of the bar it fills. Keep pouring until the prism is full.
  2. Make the height 3 cm and keep the base side 6 cm: the prism holds 6 × 6 × 3 = 108 cm³ and the pyramid 108 ÷ 3 = 36 cm³. It is still one third.
  3. Make the base side 8 cm and the height 9 cm: the prism holds 8 × 8 × 9 = 576 cm³ and the pyramid 576 ÷ 3 = 192 cm³.
  4. Make the base side 5 cm and the height 4 cm: the prism holds 5 × 5 × 4 = 100 cm³, and 100 ÷ 3 ≈ 33.3 cm³ for the pyramid. One third does not have to be a whole number.

A pyramid holds one third of the prism with the same base and height: V = ⅓ × base area × h. Here h is the perpendicular height, straight up from the middle of the base to the apex.

Why a third? A cube of side b splits into three identical pyramids that share one corner as their apex. Each has a face of the cube as its base and height b, so each is ⅓b³. Stretch the cube to any height and all three stretch together, so the third stays.

The sloping faces are triangles whose height is the slant height s, measured up the middle of a face. For a square pyramid, s² = h² + (b/2)² by Pythagoras. The whole surface area is the base plus four triangles: b² + 4 × ½ × b × s.

The height h goes in the volume. The slant height s goes in the area of the faces. They are different lengths, and s is always the longer.

base 6, height 4
5 cmslant height
48 cm³volume
96 cm²whole surface area

Base 6 cm, height 4 cm: the slant height is √(4² + 3²) = 5 cm, the volume is ⅓ × 36 × 4 = 48 cm³, and the whole surface is 36 + 60 = 96 cm².

Show me the steps
  1. Base 6 cm, height 4 cm: half the base is 3 cm, so the slant height is √(4² + 3²) = √25 = 5 cm. Volume ⅓ × 36 × 4 = 48 cm³. Faces 4 × ½ × 6 × 5 = 60 cm², surface 36 + 60 = 96 cm².
  2. Base 24 cm, height 16 cm: half the base is 12 cm, so the slant height is √(16² + 12²) = √400 = 20 cm. Volume ⅓ × 576 × 16 = 3072 cm³. Faces 4 × ½ × 24 × 20 = 960 cm², surface 576 + 960 = 1536 cm².
  3. Base 8 cm, height 3 cm: half the base is 4 cm, so the slant height is √(3² + 4²) = 5 cm. A low pyramid: volume ⅓ × 64 × 3 = 64 cm³. Faces 4 × ½ × 8 × 5 = 80 cm², surface 64 + 80 = 144 cm².
  4. Base 24 cm, height 5 cm: half the base is 12 cm, so the slant height is √(5² + 12²) = 13 cm. A very flat pyramid has a slant height much longer than its height: volume ⅓ × 576 × 5 = 960 cm³, faces 4 × ½ × 24 × 13 = 624 cm², surface 576 + 624 = 1200 cm².

A common mistake: putting the slant height into the volume, or the height into the area of a face. The height is inside the pyramid; the slant height is on its surface. If a question gives you the slant height and asks for the volume, find the height first with Pythagoras.

Worked example: a glass pyramid

A glass pyramid has a square base of side 10 m and a height of 12 m. Find the slant height, the volume and the area of glass in the four sloping faces.

  1. Half the base is 5 m. The height, half the base and the slant height make a right-angled triangle, so s² = 12² + 5² = 144 + 25 = 169, and s = 13 m.
  2. The volume is ⅓ × 10² × 12 = ⅓ × 1200 = 400 m³.
  3. Each sloping face is a triangle with base 10 m and height 13 m: ½ × 10 × 13 = 65 m². There are four, so the glass is 260 m².

Your go. The first step is given. A square pyramid has a base of side 8 cm and a height of 15 cm. Find the volume.

  1. The base area is 8 × 8 = 64 cm².
  2. Now use one third of the base area times the height.
Check the answer

The volume is ⅓ × 64 × 15 = 64 × 5 = 320 cm³. Dividing 15 by 3 first keeps the numbers small.

Check before you go on

1. A square pyramid has a base side of 6 cm and a height of 9 cm. Which calculation gives its volume?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A square pyramid has a base of side 3 cm and a height of 8 cm. What is its volume, in cm³?

3. A pyramid has a rectangular base 5 cm by 6 cm and a height of 10 cm. What is its volume, in cm³?

Level 2 · Building

4. A square pyramid has a base of side 14 cm and a height of 24 cm. What is the slant height of a sloping face, in cm?

5. A square pyramid has a base of side 20 cm and slant height 13 cm. What is its whole surface area, including the base, in cm²?

6. A pyramid has a volume of 150 cm³ and a base area of 25 cm². What is its height, in cm?

Level 3 · Stretch

7. A market stall in Lagos has a square pyramid roof with a base of side 18 m and a height of 12 m. How much canvas covers the four sloping faces, in m²?

8. A square pyramid has a base of side 30 cm and a slant height of 17 cm. What is its volume, in cm³?

Something to think about

A cube has side 6 cm. Join its centre to every corner. This cuts the cube into six identical pyramids, one on each face. What is the height of one pyramid, what is its volume, and do six of them make the cube?

One way to think about it

Each pyramid has the cube's face as its base, 36 cm², and its apex at the centre, 3 cm from the face. Its volume is ⅓ × 36 × 3 = 36 cm³. Six of them: 6 × 36 = 216 cm³, which is 6 × 6 × 6, the volume of the cube. This is the one-third rule showing up again: each pyramid is one sixth of the cube.

Exit ticket

9. A square pyramid has a base of side 20 cm and a slant height of 26 cm. What is its perpendicular height?