Geometry and measures
Combined solids
Find the surface area and volume of a solid made by joining a cylinder, a cone or a hemisphere.
A circus tent is a cone sitting on a cylinder. A capsule is a cylinder with a hemisphere at each end. An ice-cream is a cone with a hemisphere on top. None of these needs a new formula — each is two shapes you already know, joined along a circle. The only question is what happens at the join.
Volumes add. Surfaces do not.
Volume is easy: the two pieces fill separate space, so add them. Surface area is where the marks go. The circle where the pieces meet is buried inside the solid, so it belongs to neither surface — count only the faces you could actually paint.
Worked example
A toy is a cone of slant height 25 standing on a hemisphere, both of radius 7. Take π = 22/7. Find its total surface area.
Cone's curved surface: πrl = (22/7) × 7 × 25 = 550.
The slant height, not the vertical height. Using the wrong one is the single commonest slip in this topic.
A cone and a cylinder on the same base with the same height are worth knowing as a pair: the cone holds exactly a third of what the cylinder holds. It is why 1/3 appears in the cone's volume formula and nowhere in the cylinder's, and it makes a good check — if your cone comes out bigger than its cylinder, something has gone wrong.
Try it together
A solid is a cylinder of radius 7 and height 10 with a cone of radius 7 and height 12 standing on it. Take π = 22/7. Let us find its volume.
Two pieces, one addition. Do the cylinder first — it is the plainer sum.
1.The cylinder's volume is πr²h. What is (22/7) × 7 × 7 × 10?
Have a go
Have a go on your own: a cone has a radius of 7 cm and a height of 6 cm. Take π = 22/7. What is its volume, in cubic centimetres?
Hint: A third of π times the radius squared times the height.
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.