Cylinders, Cones, Spheres and Pyramids

1 · Cylinders and their nets

Recap from KS3 (Perimeter, area and volume and 3D shapes): a prism has the same cross-section all the way through, and its volume is area of cross-section × length. A cylinder is a prism with a circular cross-section, so

For the surface area, unpeel the label from a tin. It comes off as a rectangle: one side is the height h, and the other wrapped exactly once round the circle, so it's the circumference, 2πr. Add the two circular ends:

2 · Cones: slant height and curved surface

A cone has a circular base and comes to a point (the apex). Three lengths matter: the radius r, the vertical height h (apex straight down to the centre of the base) and the slant height l (apex down the side to the edge). They make a right-angled triangle, so by Pythagoras (KS3 Pythagoras):

Cut a cone up the side and flatten it: the curved surface becomes a sector of a circle with radius l. Its arc wraps round the base, so the arc is 2πr. The full circle of radius l would have circumference 2πl and area πl², so the sector is the fraction 2πr ÷ 2πl = r/l of it:

3 · Why a cone is a third of a cylinder (and a pyramid a third of a prism)

Take a cone and a cylinder with the same base and the same height. Fill the cone with water and pour it in. It takes exactly three cones to fill the cylinder. The same is true for any pyramid and the prism with the same base and height.

(Why a third? One proof: a cube can be cut into exactly three identical square-based pyramids, each with its apex at the same corner. Each is ⅓ of the cube. Squash or stretch the cube and the pieces stay equal, so ⅓ works for any pyramid.)

4 · Spheres and hemispheres

A sphere is a perfectly round ball; a hemisphere is half of one. Archimedes found the formulas by fitting a sphere snugly inside a cylinder (radius r, height 2r). The sphere's volume is exactly ⅔ of the cylinder, and its surface area equals the cylinder's curved surface. A cone in the same cylinder is ⅓, so cone + sphere = cylinder.

The exam usually gives you the sphere and cone formulas, but you must know how to use them. Don't forget the flat circular face of a hemisphere when you find its total surface area.

5 · Frustums Higher

A frustum is what's left when you slice the top off a cone with a cut parallel to the base: a lampshade or a bucket. Volume of frustum = big cone − small cone. The small cone is similar to the big one, so its radius scales with its height: r = R × h/H (see Similar shapes).

6 · Composite solids

A composite solid is made from simpler solids stuck together. Volume: add the volumes of the parts. Surface area: add only the surfaces you can see from outside; where two parts are joined, that face is hidden and doesn't count.

7 · Units: cm³, m³ and litres

Capacity is how much a container holds, measured in litres (l) and millilitres (ml). A cube 10 cm × 10 cm × 10 cm holds exactly 1 litre, so 1 litre = 1000 cm³ and 1 ml = 1 cm³. A cube 1 m on each side is 100 × 100 × 100 = 1 000 000 cm³, which is 1000 litres. Volume units jump by the cube of the length factor: 1 m = 100 cm, so 1 m³ = 100³ cm³.

8 · Practice

New question each time. Answers to 3 s.f., or as a number in front of π when asked (type 20 for 20π).

Worked examples

Quick check

What you've learned

• Cylinder: V = πr²h; curved surface 2πrh (unrolls to a 2πr × h rectangle); total 2πr² + 2πrh.
• Cone: slant height l from l² = r² + h²; curved surface πrl (unrolls to a sector); V = ⅓πr²h, a third of the cylinder.
• Pyramid: V = ⅓ × base area × height.
• Sphere: V = ⁴⁄₃πr³, S = 4πr². Hemisphere: V = ⅔πr³, total S = 3πr² (don't forget the flat face).
• Higher: frustum = big cone − small cone, using similar shapes to find the small cone.
• Composite solids: add volumes; for surface area count only the outside faces.
• 1 litre = 1000 cm³, 1 ml = 1 cm³, 1 m³ = 1 000 000 cm³ = 1000 litres.