Topic Review — Measurement: Surface Area and Volume

Mixed questions covering all three lessons. Click each answer button to reveal the solution.

1. Surface area basics. Fluency

   • (a) Rectangular prism l=8 cm, w=5 cm, h=4 cm. Find TSA.
   • (b) Cylinder r=6 cm, h=10 cm. Find TSA in terms of π.
   • (c) Sphere r=5 cm. Find TSA to 2 d.p.
   • (d) Cone r=9 cm, h=12 cm. Find TSA to 2 d.p.

2. Volume basics. Fluency

   • (a) Triangular prism: right-angle legs 6 m and 8 m, length 15 m.
   • (b) Cone r=4 cm, h=9 cm. Give answer in terms of π.
   • (c) Sphere d=14 cm. Find volume to 2 d.p.
   • (d) Square pyramid: base 10 m, height 6 m.

3. Unit conversions. Fluency

   • (a) 4.8 m² to cm².
   • (b) 750 000 mm³ to cm³.
   • (c) 2.4 m³ to litres.
   • (d) 65 000 m² to hectares.

4. Find the missing dimension. Fluency

   • (a) Cube with TSA = 216 cm². Find side length.
   • (b) Cylinder: V = 320π cm³, h = 5 cm. Find r.
   • (c) Sphere: TSA = 576π cm². Find r.
   • (d) Cone: V = 75π cm³, r = 5 cm. Find h.

5. Composite solid from a diagram. Understanding

   The diagram shows a solid made by removing a cone from a cylinder. Both have the same radius (5 cm). The cylinder is 14 cm tall; the cone has height 10 cm and sits inside, point upward, from the base.

   • (a) Write down the formula for the volume of the cylinder.
   • (b) Write down the formula for the volume of the cone.
   • (c) Find the volume of the remaining solid (cylinder minus cone) in terms of π.
   • (d) The solid is made of steel with density 7.8 g/cm³. Find its mass to the nearest gram.

6. Composite surface area. Understanding

   A trophy consists of a rectangular prism (base 8×6 cm, height 12 cm) with a square pyramid (base 8×6 cm, height 5 cm) sitting on top.

   • (a) Find the slant height of each triangular face of the pyramid. (There are two different slant heights since the base is rectangular, not square.)
   • (b) Find the total surface area of the four triangular faces of the pyramid.
   • (c) Find the total surface area of the prism (excluding the top face, since the pyramid sits there).
   • (d) Find the total surface area of the trophy.

7. Scaling and comparison. Understanding

   • (a) A sphere has volume 36π cm³. A second sphere has twice the radius. Find its volume.
   • (b) A cylinder has r=3 cm, h=8 cm. A cone has the same base and height. What percentage of the cylinder’s volume is the cone?
   • (c) A can of soup has r=3.5 cm, h=10 cm. A party size can has r=5 cm, h=15 cm. How many times more soup does the party can hold?
   • (d) All dimensions of a cone are multiplied by 1.5. By what factor does the curved surface area change?

8. Practical applications. Understanding

   • (a) A cylindrical pipe (outer r=5 cm, inner r=4 cm, length 2 m = 200 cm) is made of plastic. Find the volume of plastic used.
   • (b) A hemispherical bowl (inner radius 12 cm) is filled with water to a depth of 12 cm (full). Find the volume of water.
   • (c) Wrapping paper is cut to wrap a cylindrical gift (r=4 cm, h=20 cm). How much paper is needed if you add 10% extra for overlaps?
   • (d) A rectangular block of cheese (20×10×8 cm) is cut into cubes of side 2 cm. How many cubes are produced?

9. Water tank design. Problem Solving

   A water authority needs a tank holding exactly 10 000 L. They are considering two designs: a cylinder (r=h) and a sphere.

   • (a) For the cylindrical tank (r=h), find the radius to 2 d.p.
   • (b) Find the TSA of the cylindrical tank to 2 d.p.
   • (c) Find the radius of the spherical tank and its TSA.
   • (d) Which design uses less material (smaller TSA)? What practical factors might affect the choice?

10. Recycling an object. Problem Solving

    A solid bronze sphere of radius 9 cm is melted down and recast as a cone with base radius 6 cm.

    • (a) Find the volume of the sphere.
    • (b) Use the fact that volume is conserved to find the height of the cone.
    • (c) Find the slant height of the cone.
    • (d) By what percentage does the total surface area increase or decrease in the recast?

11. Composite solid: silo. Problem Solving

    A storage silo has a cylindrical body (r=2 m, h=6 m) with a conical base (r=2 m, h=3 m) and a hemispherical dome on top (r=2 m).

    • (a) Find the total volume of the silo.
    • (b) Find the total outer surface area (excluding the circular base of the cone, which rests on the ground).
    • (c) The silo is filled with grain (density 750 kg/m³) to 80% of its volume. Find the mass of grain.
    • (d) A new silo of the same total volume uses only a cylinder (r=2 m, no cone, no dome). Find its required height.

12. Optimisation. Problem Solving

    A manufacturer wants to make a closed cylindrical tin with a fixed volume of 1000 cm³. The cost of the material is $0.02 per cm².

    • (a) Express the height h in terms of r using the volume formula.
    • (b) Express the total surface area as a function of r alone.
    • (c) By testing r = 3, 4, 5, 6, 7 cm, estimate the value of r that minimises surface area.
    • (d) At the optimal radius found in (c), find the manufacturing cost per tin.