We have a cylinder with a base radius of r cm and height h cm. A square pyramid is fitted inside it. The square base of the pyramid lies on the base of the cylinder, its corners on the boundary of the cylinder. The apex of the pyramid lies on the top of the cylinder. Find the ratio of the volume of this pyramid to the volume of the cylinder.

Step-by-step solution

Idea: Volume of a pyramid = 13 × area of base × height. The square’s corners lie on the circle, so its diagonal is a diameter.

1. The square’s diagonal = diameter = 2r. Area of a square = (diagonal)²2 = 4r²2 = 2r². (Side = r√2.)1 mark
2. Height of pyramid = h. Volume = 13 × 2r² × h = 2r²h3.1 mark
3. pyramidcylinder = 2r²h/3πr²h = 23π. Ratio = 2 : 3π ≈ 0.212 : 1.1 mark

Check: The pyramid is inside the cone with the same base circle and height, whose ratio is 13 ≈ 0.33; 0.21 < 0.33 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Taking the square’s side as 2r (the diameter) instead of its diagonal.
• Forgetting the 13 in the pyramid formula.

How this can come in the exam

Try one yourself

A square pyramid of base side 6 cm and height 10 cm: find its volume.

More questions like this

• What is the change in volume when:
• Given a cube with volume V, express its total surface area S in terms of V.
• Given a cube with total surface area S, express its volume V in terms of S.
• A cylindrical glass of height 25 cm and radius 4 cm has water up to a height of 16 cm. A crow wants to drink water from this glass. The water must be at a height of 20 cm for the crow to reach it. There are some marbles lying around. How many marbles of radius 1 cm should the crow drop into the glass to make the water reach the required height?
• Find the volume of ink in a new ball point pen. Take the necessary measurements and make approximations, as needed.

All Surface area and volume questions · All maths questions