M4.5.2 Chapter 2

Pyramids and cones

Does a cone hold as much as the cylinder it fits inside?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A Half as much as the cylinder.
You picked A. Read on. At the end, the chapter comes back to your guess.
B The same amount as the cylinder.
You picked B. Read on. At the end, the chapter comes back to your guess.
C One third as much as the cylinder.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Find the volume of a pyramid as 1/3 × base area × height
Find the volume and curved surface area of a cone

A pyramid and a cone both rise from a flat base to a single point, the apex. A tent, a funnel and an ice cream cone all have this shape.

Take a pyramid and a prism with the same base and the same height. The pyramid fits inside the prism. How much of the prism does it fill?

Draw it first

A pyramid inside a prism

1 Draw a pyramid with a square base. Its height is the straight line from the apex down to the base.
2 Draw the box with the same base and the same height round the pyramid.
3 Three pyramids like this one, poured as sand, fill the box exactly once.

This is a fact you can test with sand, and it can be proved in more advanced mathematics. The pyramid fills one third of the box. The box has volume base area × height. So for a pyramid, V = ⅓ × base area × height.

The height must be the perpendicular height. It is measured at a right angle to the base, straight down from the apex. The slant height runs down a sloping face. It is longer, and it is not used in this volume.

The perpendicular height is 4 cm. The slant height of the front face is 5 cm.

Worked example

Mei builds a model pyramid with a square base of side 6 cm. Its perpendicular height is 4 cm. Find its volume.

The base is a square. Its area is 6 × 6 = 36 cm².
Volume = ⅓ × base area × perpendicular height = ⅓ × 36 × 4.
⅓ × 36 × 4 = 12 × 4 = 48 cm³.

Answer: 48 cm³

A cone is a pyramid with a circle for its base. The base area is πr², so the volume of a cone is V = ⅓πr²h, with h the perpendicular height.

The cone fills one third of the cylinder round it.

Worked example

A cone has base radius 3 cm and perpendicular height 4 cm. Find its volume, to 1 decimal place.

The base is a circle of radius 3 cm. Its area is π × 3² = 9π cm².
Volume = ⅓ × base area × height = ⅓ × 9π × 4.
⅓ × 9π × 4 = 3π × 4 = 12π cm³.
Put in the π key of a calculator. 12π is 37.7 cm³, to 1 decimal place.

Answer: 37.7 cm³

Now the curved surface of a cone. Cut it open and lay it flat. You get a sector of a circle, with a circle for the base beside it.

The sector has radius l, the slant height. Its curved edge is the circumference of the base.

The straight sides of the sector are the slant height l. The curved edge is the circumference of the base circle, 2πr. A whole circle of radius l has circumference 2πl and area πl².

Our sector is the fraction 2πr ÷ 2πl = r ÷ l of that whole circle. So its area is (r ÷ l) × πl² = πrl. The curved surface area of a cone is πrl.

To find l, use Pythagoras on the right angled triangle inside the cone. Its sides are r, the perpendicular height h, and l: l² = r² + h². So l = √(r² + h²).

Worked example

A cone has base radius 3 cm and perpendicular height 4 cm. Find the area of its curved surface, to 1 decimal place.

The slant height l joins the apex to the rim. It is the longest side of a right angled triangle with sides 3 and 4.
l² = 3² + 4² = 9 + 16 = 25.
l = √25 = 5 cm.
Curved surface area = π × r × l = π × 3 × 5 = 15π cm².
Put in the π key of a calculator. 15π is 47.1 cm², to 1 decimal place.

Answer: 47.1 cm²

Try it

Three cones fill a cylinder

You need: a sheet of stiff paper, a piece of tape, dry rice, sand or salt.

Roll the paper into a cone and tape it. Cut the wide end flat, so the cone can stand on it.
Roll a second sheet into a cylinder with the same wide end and the same height.
Fill the cone with rice and tip it into the cylinder.
Count how many cones of rice fill the cylinder.

What you will see: You need three cones. The cone holds one third as much as the cylinder, which is why there is a ⅓ in the volume of a cone.

If you cannot try it, this is what happens.

A wrong idea many people have

In the volume of a cone, h is the slant height.

Why it feels right: The slant height is the length you can see along the side, and you can measure it with a ruler.

The test that breaks it: Take the cone with radius 3 cm, perpendicular height 4 cm and slant height 5 cm. The true volume is ⅓ × π × 9 × 4 = 12π, which is 37.7 cm³ to 1 decimal place. With 5 you get ⅓ × π × 9 × 5 = 15π, which is 47.1 cm³, and that is too big.

What is true: Use the perpendicular height, at a right angle to the base. The slant height belongs in the curved surface, πrl.

A wrong idea many people have

The volume of a pyramid is base area × height, like a prism.

Why it feels right: A pyramid has a base and a height, and for prisms we multiply them.

The test that breaks it: Take the pyramid with a 6 cm square base and a height of 4 cm. Base area × height is 36 × 4 = 144 cm³. That is the whole box. Three pyramids fill the box, so one pyramid is 144 ÷ 3 = 48 cm³.

What is true: A pyramid fills only one third of the prism with the same base and height. Multiply by ⅓.

Key words

apex: the point at the top of a pyramid or a cone.
perpendicular height: the height from the apex straight down to the base, at a right angle to it.
slant height: the length from the apex down the sloping side to the base edge.

Back to your guess

The answer is C: One third as much as the cylinder.

A cone with the same base and height as a cylinder holds exactly one third as much. Fill the cone with rice, tip it into the cylinder, and you need three cones to fill it.

Keep it

Copy this into your exercise book.

Pyramid and cone: V = ⅓ × base area × perpendicular height. Cone V = ⅓πr²h. Curved surface πrl, with l² = r² + h². Not the slant height in V.

Draw: A cone with r, h and l marked, the height as a dashed upright line with a right angle at the base.