Practice Maths

Surface Area of Prisms and Cylinders

Key Ideas

Key Terms

surface area (SA)
Of a solid is the total area of all its faces (or curved surface).
net
A 2D flat layout of all faces — useful for visualising and calculating SA.
Rectangular prism
SA = 2(lw + lh + wh), where l = length, w = width, h = height.
Triangular prism
SA = 2 × (area of triangular face) + sum of the three rectangular face areas.
Cylinder
SA = 2πr² + 2πrh, where r = radius and h = height. The two circles are the top and bottom; 2πrh is the curved surface (rectangle when unrolled).
Hot Tip For a cylinder, the curved surface area unrolls into a rectangle with width 2πr (the circumference) and height h. So curved SA = 2πrh. Add the two circular ends: 2πr².

Worked Example

Question: Find the total surface area of a cylinder with radius r = 5 cm and height h = 12 cm. Give the exact answer and an approximation to 1 decimal place.

Step 1 — Two circular ends:   2πr² = 2 × π × 5² = 2 × 25π = 50π cm²

Step 2 — Curved surface:   2πrh = 2 × π × 5 × 12 = 120π cm²

Step 3 — Total SA:   SA = 50π + 120π = 170π ≈ 534.1 cm²

What is Surface Area?

The surface area of a solid is the total area of all its outer faces — the amount of material you would need to wrap around the outside. Surface area is measured in square units: cm2, m2, etc. It is used in real life for things like calculating how much paint to buy, how much cardboard is needed for a box, or how much insulation wraps a pipe.

Surface Area of a Rectangular Prism

A rectangular prism (box) has 6 rectangular faces that come in 3 matched pairs. If the dimensions are length (l), breadth (b), and height (h):

SA = 2(lb + lh + bh)

Each term covers one pair of opposite faces. For example, lb is the area of the top and bottom, lh is the front and back, bh is the two sides.

Useful check: sketch the net (unfolded shape) of the prism and add up each rectangle's area separately — you should get the same total.

Surface Area of a Cylinder

A cylinder has three faces: two circular ends (top and bottom) plus one curved rectangular surface (the "label" that wraps around). When you unroll the curved surface it becomes a rectangle with width = height of the cylinder (h) and length = circumference of the circle (2πr).

SA = 2πr2 + 2πrh

  • 2πr2 accounts for both circles (each has area πr2)
  • 2πrh accounts for the curved side surface

If the cylinder is open at one or both ends (like a pipe), subtract the missing circular face(s). Always read the question carefully.

Using Nets to Visualise Surface Area

A net is the 2D shape you get when you unfold a solid completely flat. Drawing a net is a powerful strategy: it forces you to count every face, and you can label and calculate each face's area before adding them all up. Nets prevent the common mistake of forgetting a face or counting one twice.

Common mistake: Forgetting that a cylinder's curved surface is a rectangle when unrolled, not a circle. The width of that rectangle is the circumference (2πr), not the radius. Also, always include units2 in your final answer — surface area is always in square units. Writing SA = 150 without cm2 will cost you marks.

Mastery Practice

  1. Find the total surface area of each rectangular prism. Fluency

    1. l = 8 cm, w = 5 cm, h = 3 cm
    2. l = 10 m, w = 4 m, h = 6 m
    3. l = 7.5 cm, w = 7.5 cm, h = 7.5 cm (a cube)
    4. l = 12 mm, w = 9 mm, h = 2 mm
    5. l = 1.2 m, w = 0.8 m, h = 0.5 m

  2. Find the total surface area of each cylinder. Give your answer exact (in terms of π) and rounded to 1 decimal place. Fluency

    1. r = 3 cm, h = 10 cm
    2. r = 7 m, h = 2 m
    3. r = 4.5 cm, h = 8 cm
    4. r = 1.5 m, h = 5 m
    5. diameter = 20 cm, h = 15 cm

  3. Find the total surface area of each triangular prism. Fluency

    1. The triangular face is a right triangle with legs 6 cm and 8 cm (hypotenuse 10 cm). The prism length is 15 cm.
    2. The triangular face is equilateral with side 6 m. The prism length is 10 m. (Use area = (√3/4) × 6².)
    3. The triangular face has base 9 cm, perpendicular height 4 cm, and the three sides are 9 cm, 5 cm, and 5 cm. The prism length is 12 cm.

  4. A cylinder has a radius of 6 cm and a height of 10 cm. Understanding

    1. Sketch the net of the cylinder, labelling all dimensions.
    2. Calculate the area of each circular face.
    3. Calculate the area of the curved surface (the rectangle in the net).
    4. Hence find the total surface area, to 2 decimal places.
    5. If the cylinder were open at one end (like a cup), what would its surface area be?

  5. Compare the surface areas of these two solids. Understanding

    1. Solid A: a rectangular prism with l = 10 cm, w = 10 cm, h = 10 cm.
    2. Solid B: a cylinder with diameter = 10 cm and h = 10 cm.
    3. Which solid has the greater surface area? By how much (to 1 d.p.)?
    4. A student says “a cube always has more surface area than a cylinder of the same height.” Use your results to comment on this claim.

  6. Find the missing dimension in each case. Understanding

    1. A cylinder has SA = 100π cm² and radius = 5 cm. Find its height.
    2. A cube has total SA = 294 cm². Find the side length.
    3. A rectangular prism has l = 8 cm, w = 3 cm, and SA = 158 cm². Find h.

  7. Unit conversion and surface area. Understanding

    1. A rectangular prism has dimensions 2 m × 1.5 m × 0.8 m. Calculate the SA in m², then convert your answer to cm².
    2. A cylindrical pipe has radius 15 mm and length 2 m. Convert all dimensions to centimetres, then find the curved surface area only (the pipe has open ends). Give your answer in cm².
    3. 1 m² = how many cm²? Use this fact to verify part (a).

  8. A storage shed is in the shape of a triangular prism. Problem Solving

    The two triangular ends are isosceles triangles with base 6 m and slant sides 5 m each (perpendicular height of triangle = 4 m). The shed is 10 m long. The base of the shed sits on the ground and is not painted.

    1. Find the area of one triangular end.
    2. Find the area of the two rectangular sloping roof panels combined.
    3. Find the total area to be painted (two triangular ends + two roof panels; the base is excluded).
    4. Paint covers 12 m² per litre. How many litres are needed (round up to the next whole litre)?
    5. Paint costs $18.50 per litre. What is the total cost?

  9. A tin can (closed cylinder) has a diameter of 8 cm and a height of 11 cm. Problem Solving

    1. Find the total surface area of the can to 2 decimal places.
    2. The manufacturer wants to make a label that wraps around the curved surface only, with a 0.5 cm overlap along the height. Find the dimensions of the label (width and length).
    3. If the label material costs $0.003 per cm², find the cost of one label to the nearest cent.
    4. Find the cost of labelling 10 000 cans.

  10. A solid is made by placing a cylinder (r = 3 cm, h = 4 cm) on top of a rectangular prism (8 cm × 8 cm × 5 cm). The cylinder sits centrally on the top face of the prism. Problem Solving

    1. Draw and label a sketch of this composite solid.
    2. Find the surface area of the rectangular prism, excluding the circular region where the cylinder sits.
    3. Find the surface area of the cylinder, excluding its bottom circular face (which sits on the prism).
    4. Hence find the total external surface area of the composite solid (to 1 d.p.).
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