Mensuration
Surface area and volume
Find surface areas and volumes of cones, spheres and hemispheres.
Cuboids and cylinders were straightforward: a flat face, carried straight upwards. Cones, spheres and hemispheres are not, because nothing about them is straight. The formulas below are the ones that let you handle them anyway, and they are worth knowing by heart — but each of the awkward ones hides a right-angled triangle or a neat fraction that makes it easier to remember than it looks.
The slant height is a hypotenuse
A cone has three lengths and they are not independent. The radius r and the height h meet at a right angle inside the cone, and the slant height l runs up the outside from the rim to the tip. So l² = r² + h², and the curved surface area π × r × l always wants l, never h.
Worked example
A cone has a radius of 6 units and a height of 8 units. Find its slant height.
Square the two known lengths: 6 × 6 = 36 and 8 × 8 = 64.
The radius and the height are the two short sides of the right-angled triangle inside the cone.
Try it together
A cone has a radius of 7 units and a height of 24 units. Find its slant height, then its volume with π = 22/7.
Two steps of Pythagoras, then one formula. The numbers have been chosen so nothing awkward turns up.
1.Square the radius: what is 7 × 7?
Watch out
A hemisphere has two different surface areas. Its curved surface alone is 2 × π × r × r, but the total surface adds the flat circular lid, π × r × r, giving 3 × π × r × r. Read which one the question wants before working anything out.
Have a go
Have a go on your own: which of these gives the surface area of a sphere?
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.