Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
Level 1 · Warm-up
1 Find the surface area of this closed cylinder.
Show worked solution
Circles: 2 × 49π = 98π. Curved: 2π × 7 × 3 = 42π. Total 140π cm2. 42π cm2 is the curved surface alone; 91π cm2 has one circle; 147π cm2 is the volume πr2h.
2 A tin with no lid has a diameter of 18 cm and a height of 4 cm. Find the area of its base, its curved surface, and the total.
Show worked solution
r = 9. Base: π × 81 = 81π. Curved: 2π × 9 × 4 = 72π. Total: 153π cm2.
3 A pipe open at both ends has radius 3 cm and an outside surface area of 84π cm2. How long is it, in cm?
Show worked solution
2π × 3 × L = 6πL = 84π, so L = 84 ÷ 6 = 14.
Level 2 · Core
4 This solid is half of a cylinder, cut along its length. The radius is 4 cm and the length is 10 cm. Find its total surface area.
Show worked solution
Two half-circles: π × 42 = 16π. Half the curved surface: ½ × 2π × 4 × 10 = 40π. The flat base is the diameter × the length: 8 × 10 = 80. Total 80 + 56π cm2. 56π cm2 forgets the flat face; 40 + 56π makes the flat face 4 wide instead of 8; 80 + 72π uses two whole circles for the ends.
5 A 4 cm cube is glued to the middle of the top of an 8 cm × 8 cm × 2 cm slab.Find the total surface area of the two solids separately, the area hidden by the join, and the surface area of the glued solid.
Show worked solution
Slab 192 + cube 96 = 288. Hidden: two 4 × 4 faces = 32. Glued solid: 288 − 32 = 256 cm2.
6 Three 2 cm cubes are glued face to face in a straight row. Find the surface area of the row, in cm2.
Show worked solution
Three cubes: 3 × 24 = 72. Two joins each hide two 4 cm2 faces: 2 × 8 = 16. SA = 72 − 16 = 56. Check: the row is a 6 × 2 × 2 box, 2(12 + 12 + 4) = 56 ✓.
Level 3 · Challenge
7 Three statements:
1 A closed cylinder whose height equals its radius r has surface area 4πr2.
2 Two closed cylinders with r = 2 cm and h = 5 cm, glued end to end, make a solid with surface area 56π cm2.
3 A tin with no lid, r = 3 cm and h = 6 cm, uses 45π cm2 of metal.
Which statements are correct?
1 A closed cylinder whose height equals its radius r has surface area 4πr2.
2 Two closed cylinders with r = 2 cm and h = 5 cm, glued end to end, make a solid with surface area 56π cm2.
3 A tin with no lid, r = 3 cm and h = 6 cm, uses 45π cm2 of metal.
Which statements are correct?
Show worked solution
2πr2 + 2πr × r = 4πr2 (1 ✓). Each cylinder is 8π + 20π = 28π, but gluing hides two 4π circles: 56π − 8π = 48π, not 56π (2 ✗). 9π + 2π × 3 × 6 = 9π + 36π = 45π (3 ✓). 1 and 3.
8 A closed cylinder has radius 8 cm and a surface area of 288π cm2. Find its height and its curved surface area.
Show worked solution
Circles: 128π. Curved: 288π − 128π = 160π = 16πh, so h = 10 cm.
9 This L-shaped prism is 4 cm deep. Its front is a 6 cm square with a 3 cm square cut from one corner. Find its surface area, in cm2.
Show worked solution
L area: 36 − 9 = 27, so front and back give 54. The perimeter of the L is 6 + 6 + 3 + 3 + 3 + 3 = 24, so the faces around it give 24 × 4 = 96. SA = 54 + 96 = 150.