Surface Area of a Prism
The surface area of a prism is the total area of all its faces, which is easiest to see on its net. This NSW Year 9 Advanced maths subtopic finds surface areas of rectangular, triangular and composite prisms (NESA outcome MA5‑ARE‑C‑01).
You will find surface areas of closed and open boxes, triangular prisms and prisms with L-shaped cross-sections, using Pythagoras' theorem for sloping faces. You will cost painting a crate and tiling a pool, find the metal for a trough, find a cube's edge or a prism's length from its surface area, and see why doubling every edge multiplies the surface area by \(4\). It is part of Year 9 Advanced Surface Area and Volume.
Theory
The surface area of a prism is the total area of all its faces: two identical ends plus one rectangle for every side of the end. This Year 9 Advanced maths page covers nets, open and floorless shapes, sloping faces, costs and working backwards to an unknown edge.
A right prism has two identical, parallel ends (the cross-section) joined by rectangles at right angles to them. It is named after its cross-section: rectangular, triangular, trapezoidal.
The surface area (SA) is the total area of all the faces. A net is the shape you get by unfolding the prism flat; its area is the surface area.
A prism whose end has \(n\) sides has \(n + 2\) faces: the two ends and \(n\) rectangles. Each rectangle is one side of the end by the length of the prism.
Real objects are often missing faces: an open box has no lid, a shed may have no floor, a trough has no top. List only the faces that are there.
Rectangular prism (closed):
Any right prism with end area \(A\), end perimeter \(P\) and length \(L\):
How to find the surface area of a prism
- Sketch or unfold the prism and identify the cross-section.
- List the faces: two ends and one rectangle per side of the end. Cross out any face that is not there (no lid, no floor).
- Find missing lengths, such as a sloping edge, with Pythagoras' theorem.
- Find each area: the end shape with its own formula (triangle, trapezium, composite), each rectangle as side \(\times\) length.
- Add and give square units. For a cost, convert to the units of the price first.
Harder problems
- Composite ends: for an L-shaped or stepped cross-section, find the end area by splitting, and the rectangles by perimeter \(\times\) length.
- Working backwards: write the surface area with the unknown, set it equal to the given value and solve. A quadratic may appear; reject a negative length.
- Joined solids: when cubes or boxes are glued face to face, the hidden faces are no longer on the surface, so subtract them (two per join).
- Patterns: for a row of \(n\) identical blocks, count the faces that grow with \(n\) and the faces that do not, and write SA as an expression in \(n\).
The sloping edge is the hypotenuse of a right-angled triangle with legs \(1.5\) m (half the width) and \(2\) m.
| \(s\) | \(=\) | \(\sqrt{1.5^2 + 2^2} = \sqrt{6.25} = 2.5~\text{m}\) |
List the five faces:
| \(\text{two ends}\) | \(=\) | \(2 \times \dfrac{1}{2} \times 3 \times 2 = 6\) |
| \(\text{two sloping sides}\) | \(=\) | \(2 \times 2.5 \times 4 = 20\) |
| \(\text{floor}\) | \(=\) | \(3 \times 4 = 12\) |
| \(\text{SA}\) | \(=\) | \(6 + 20 + 12 = \mathbf{38}~\text{m}^2\) |
The rectangles around a prism, laid side by side, make one long rectangle: the perimeter of the end times the length. Find the slanted top edge first.
| \(\text{top}\) | \(=\) | \(\sqrt{2.4^2 + 0.6^2} = 2.4738\ldots~\text{m}\) |
| \(\text{one end}\) | \(=\) | \(\dfrac{2.4}{2}(0.4 + 1.0) = 1.68~\text{m}^2\) |
| \(\text{perimeter}\) | \(=\) | \(0.4 + 1.0 + 2.4 + 2.4738\ldots = 6.2738\ldots~\text{m}\) |
| \(\text{SA}\) | \(=\) | \(2 \times 1.68 + 6.2738\ldots \times 1.5\) |
| \(\) | \(=\) | \(3.36 + 9.4108\ldots \approx \mathbf{12.77}~\text{m}^2\) |
No lid means five faces: the base, two long sides and two short ends.
| \(\text{base}\) | \(=\) | \(80 \times 35 = 2800\) |
| \(\text{long sides}\) | \(=\) | \(2 \times 80 \times 45 = 7200\) |
| \(\text{ends}\) | \(=\) | \(2 \times 35 \times 45 = 3150\) |
| \(\text{SA}\) | \(=\) | \(13\,150~\text{cm}^2\) |
The price is per square metre, and \(1~\text{m}^2 = 10\,000~\text{cm}^2\).
| \(13\,150 \div 10\,000\) | \(=\) | \(1.315~\text{m}^2\) |
| \(\text{cost}\) | \(=\) | \(1.315 \times 95 = 124.925 \approx \mathbf{\$124.93}\) |
(i) Show that \(x^2 + 18x - 144 = 0\).
(ii) Find the side of the base.
(i) Two square faces (top and base) and four rectangles \(x\) by \(9\).
| \(2x^2 + 4 \times 9x\) | \(=\) | \(288\) |
| \(2x^2 + 36x - 288\) | \(=\) | \(0\) |
| \(x^2 + 18x - 144\) | \(=\) | \(0\) |
(ii) Factorise: two numbers with product \(-144\) and sum \(18\) are \(24\) and \(-6\).
| \((x + 24)(x - 6)\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(\mathbf{6}~\text{cm} \quad (x = -24 \text{ rejected})\) |
Check: \(2 \times 36 + 36 \times 6\) \(= 72 + 216\) \(= 288\) \(\checkmark\)
Common pitfalls
Frequently asked questions
How do you find the surface area of a prism?
Add the areas of all its faces: the two identical ends and one rectangle for each side of the end. Each rectangle is a side of the end times the length of the prism.
What is the formula for the surface area of a rectangular prism?
\(\text{SA} = 2(lw + lh + wh)\). For a cube of side \(s\) it is \(6s^2\).
How do you find the surface area of a triangular prism?
Add the two triangular ends, each \(\dfrac{1}{2} \times\) base \(\times\) height, and the three rectangles, each a side of the triangle times the length of the prism. Use Pythagoras' theorem if a side of the triangle is missing.
What is the surface area of an open box?
Leave out the missing lid: add the base, the two long sides and the two short ends, five faces in total.
Is there a shortcut for the surface area of any prism?
Yes. \(\text{SA} = 2A + PL\): twice the area of the end plus the perimeter of the end times the length.
How do you find the edge of a cube from its surface area?
Divide the surface area by \(6\) to get the area of one face, then take the square root.