Surface Areas and Volumes

1. What this chapter is about

In earlier classes you found the surface area and volume of cuboids, cubes and cylinders. This chapter completes the family by adding the right circular cone, the sphere and the hemisphere. For each solid you must know two things: its surface area (the size of its outer covering) and its volume (how much it can hold). Surface area is always in square units, and volume is always in cubic units — getting the units right is half the battle in this chapter.

2. The right circular cone

Imagine an ice-cream cone or a birthday hat. It stands on a circular base of radius r. The pointed top is called the apex. The straight distance from the apex down to the centre of the base is the height h. The distance from the apex to any point on the edge (rim) of the base is the slant height l. These three are connected by the Pythagoras theorem, because the radius, the height and the slant height form a right-angled triangle: l² = r² + h², so l = √(r² + h²).

The curved (lateral) surface area of a cone — the slanting part only — is πrl. If you also cover the circular base, you get the total surface area = curved surface + base = πrl + πr² = πr(l + r). The volume of a cone is exactly one-third of the volume of a cylinder with the same base and height: V = (1/3)πr²h. A handy way to remember this: three cones full of water exactly fill one cylinder of the same radius and height.

3. The sphere

A sphere is a perfectly round solid like a football, a marble or a planet. Every point on its surface is the same distance r (the radius) from its centre. A sphere has only one kind of area — there is no flat base — so its surface area is simply 4πr². An interesting fact is that this is exactly the area of four circles of the same radius. The volume of a sphere is (4/3)πr³. Notice the radius is cubed for volume (a measure of space) and squared for surface area (a measure of skin).

4. The hemisphere

Cut a sphere exactly in half through its centre and you get a hemisphere (think of half an orange or a bowl). A hemisphere has two parts to its surface: the curved part (the dome) and the new flat circular face created by the cut. The curved surface area is half the sphere’s surface = 2πr². The total surface area adds the flat circle (πr²) to give 2πr² + πr² = 3πr². Be careful: use curved surface area for an open bowl, but total surface area for a solid hemisphere. The volume is half the sphere’s volume = (2/3)πr³.

5. Choosing curved vs total surface area

Read the question carefully. If a cone is open at the base (like a clown’s hat or a funnel), use the curved surface area πrl only. If it is a solid cone or has a base lid, use the total surface area πr(l + r). The same logic applies to a hemisphere: an open bowl uses 2πr², a solid dome uses 3πr². The words “open”, “closed”, “solid” and “hollow” are the clues.

6. Working with π, units and accuracy

Use π = 22/7 when the radius is a multiple of 7 (it cancels neatly); otherwise use π = 3.14. Always keep every measurement in the same unit before substituting — convert metres to centimetres or vice versa first. Express your final surface-area answer in square units and your volume answer in cubic units. For capacity of liquids remember 1 litre = 1000 cm³ and 1 m³ = 1000 litres.

7. A quick comparison

For solids of the same radius and height: a cylinder holds the most, a cone holds the least (one-third of the cylinder), and a sphere/hemisphere sit in between depending on the dimensions. Spotting these relationships helps you check whether an answer is sensible. If your cone volume ever comes out bigger than the matching cylinder, you have made a mistake!