Practice Maths

L19 — Surface Area of Prisms and Cylinders

Key Terms

Total surface area (TSA)
The sum of the areas of all faces of a 3D solid, measured in square units.
Prism
A solid with two identical parallel ends (cross-sections) and rectangular side faces.
Cylinder
A prism with circular ends. Curved surface unrolls into a rectangle: width = 2πr, height = h.
Net
The 2D shape formed by unfolding all faces of a solid flat — useful for visualising and calculating TSA.
Slant height (l)
For a cone: the distance from the apex to the base edge along the surface. l = √(r² + h²).
Open container
A solid with one face missing (e.g. no lid). Simply omit that face from the TSA calculation.

Total Surface Area (TSA)

The total surface area is the sum of the areas of all faces of a solid.

l h w
Rectangular prism with length l, width w, height h.
ShapeTSA Formula
Rectangular prism 2(lw + lh + wh)
Cylinder (closed) 2πr² + 2πrh
Triangular prism 2 × (triangle area) + sum of 3 rectangle areas
Sphere 4πr²
Cone (closed base) πr² + πrl  (l = slant height)

For a cylinder: the curved surface “unrolls” into a rectangle of width 2πr and height h, giving area 2πrh.

Hot Tip: Draw a net for unfamiliar shapes — a cylinder becomes two circles and a rectangle. For composite solids, don’t count the contact area between joined shapes: it’s hidden inside and not part of the surface.

Rectangular Prism (Cuboid)

A rectangular prism has 6 rectangular faces in 3 matching pairs. Opposite faces are identical.

Worked Example 1 — TSA of a rectangular prism

Find the TSA of a box with l = 8 cm, w = 5 cm, h = 3 cm.

TSA = 2(lw + lh + wh) = 2(8×5 + 8×3 + 5×3) = 2(40 + 24 + 15) = 2 × 79 = 158 cm²

Cylinder

A closed cylinder has two circular ends and one curved surface. The curved surface unrolls into a rectangle: width = circumference = 2πr, height = h.

Worked Example 2 — TSA of a cylinder

Find the TSA of a cylinder with radius 4 cm and height 10 cm. (Give answer to 2 d.p.)

TSA = 2πr² + 2πrh = 2π(4)² + 2π(4)(10) = 32π + 80π = 112π ≈ 351.86 cm²

Triangular Prism

A triangular prism has two triangular ends and three rectangular side faces. Identify each rectangular face and its dimensions carefully.

Worked Example 3 — TSA of a triangular prism

A triangular prism has a right-triangle cross-section with legs 3 cm and 4 cm (hypotenuse 5 cm). The prism is 8 cm long.

Two triangular ends: 2 × (½ × 3 × 4) = 12 cm²

Three rectangles: (3×8) + (4×8) + (5×8) = 24 + 32 + 40 = 96 cm²

TSA = 12 + 96 = 108 cm²

Open Containers

If a shape has no lid (open top), simply omit that face from the calculation.

  1. TSA of rectangular prisms. Fluency

    Give answers to 2 decimal places where not exact.

    • (a) l = 6 cm, w = 4 cm, h = 3 cm.
    • (b) l = 10 m, w = 7 m, h = 2 m.
    • (c) A cube with side 5 cm.
    • (d) l = 12 mm, w = 8 mm, h = 5 mm.

  2. TSA of cylinders. Fluency

    Give answers in terms of π and as a decimal (2 d.p.).

    • (a) r = 3 cm, h = 8 cm.
    • (b) r = 5 m, h = 12 m.
    • (c) Diameter = 10 cm, h = 6 cm.
    • (d) r = 2.5 mm, h = 9 mm.

  3. TSA of triangular prisms. Fluency

    Give answers to 2 decimal places.

    • (a) Right-triangle cross-section: legs 5 cm and 12 cm (hypotenuse 13 cm). Length 10 cm.
    • (b) Equilateral triangle cross-section: side 6 cm. Length 8 cm.
    • (c) Right-triangle cross-section: legs 8 m and 15 m (hypotenuse 17 m). Length 20 m.
    • (d) Isosceles triangle cross-section: base 8 cm, equal sides 5 cm, height 3 cm. Length 15 cm.

  4. Find the missing dimension. Fluency

    • (a) Rectangular prism: TSA = 148 cm², l = 6 cm, w = 4 cm. Find h.
    • (b) Cylinder: TSA = 54π cm², r = 3 cm. Find h.
    • (c) Cube: TSA = 294 cm². Find the side length.
    • (d) Cylinder: TSA = 100π m², h = 8 m. Find r.

  5. Surface area from the diagram. Understanding

    The diagram shows a triangular prism. The cross-section is a right-angled triangle.

    6 cm 8 cm 10 cm 12 cm
    • (a) How many faces does this prism have? List them by shape and dimensions.
    • (b) Find the area of each triangular face.
    • (c) Find the area of each rectangular face (three rectangles).
    • (d) Find the total surface area.

  6. Open containers and partial surfaces. Understanding

    • (a) An open-top rectangular box: l = 40 cm, w = 30 cm, h = 20 cm. Find the surface area of the inside (base + four walls).
    • (b) An open-top cylinder (no lid): r = 6 cm, h = 10 cm. Find the surface area to 2 d.p.
    • (c) A half-cylinder (split lengthwise): r = 5 cm, h = 12 cm. Find the total surface area. (Hint: include the flat rectangular base and the two semicircular ends.)
    • (d) A cylinder is open at both ends (a tube): r = 4 cm, h = 15 cm. Find the curved surface area.

  7. Composite shapes. Understanding

    • (a) A solid is made by placing a small cube (side 3 cm) on top of a large cube (side 8 cm). Find the total exposed surface area. (The contact area is not exposed.)
    • (b) A rectangular prism (10 × 6 × 4 cm) has a cylindrical hole of radius 1 cm drilled straight through its 10 cm length. Find the total surface area of the resulting solid. (Give your answer to 2 d.p.)
    • (c) A shed roof is a triangular prism (equilateral triangle, side 4 m, length 10 m) sitting on top of a rectangular prism (4 × 10 × 3 m). What is the total outer surface area? (The roof sits flush on the walls — do not include the contact area.)
    • (d) A cylinder (r = 3 cm, h = 8 cm) sits on top of a rectangular prism (10 × 10 × 5 cm). Find the total surface area. (The cylinder’s base circle sits on top of the prism.)

  8. Spheres and cones. Understanding

    Use: sphere TSA = 4πr²; cone TSA = πr² + πrl (l = slant height = √(r² + h²)).

    • (a) Sphere: r = 7 cm. Find the TSA in terms of π and as a decimal.
    • (b) Sphere: TSA = 324π cm². Find the radius.
    • (c) Cone: r = 6 cm, h = 8 cm. Find the slant height, then the TSA.
    • (d) An ice-cream cone (open top, no base circle) has r = 4 cm and l = 11 cm. Find the curved surface area.

  9. Painting and wrapping. Problem Solving

    A garden shed is a rectangular prism (5 m × 3 m base, 2.4 m walls) with a triangular-prism roof (isosceles triangle: base 3 m, slant sides 2 m, running the full 5 m length of the shed). The floor is not painted. The two triangular gable ends and the two sloping roof surfaces are painted a different colour from the four walls.

    • (a) Find the area of one gable-end triangle. (Use Pythagoras to find the roof height first.)
    • (b) Find the total area of the four walls.
    • (c) Find the total area of the two sloping roof surfaces.
    • (d) Paint costs $12/m² for roof and $8/m² for walls. Find the total painting cost.

  10. Packaging design. Problem Solving

    A company packages a cylindrical canister (r = 5 cm, h = 18 cm) inside a rectangular box that fits it exactly (the cylinder just touches all four sides of the box, and the heights are equal).

    • (a) What are the dimensions of the rectangular box?
    • (b) Find the TSA of the cylindrical canister to 2 d.p.
    • (c) Find the TSA of the rectangular box.
    • (d) How much more cardboard (TSA) does the box use than the canister? Give the answer as a percentage.
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