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This page sets out the method for surface area in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Surface area topic page.
For any solid with flat faces, make a table of faces before calculating.
For a cuboid, use the three different faces and double each.
Surface area adds the areas of the faces.
It is not the volume, and it is measured in square units.
The three different faces are length × width, width × height, and height × length.
Double that sum, because each face has a matching one opposite.
A net is the solid folded out flat.
The surface area is the area of the net. Do not count a face twice where the net only shows it once.
Two end faces, plus a rectangle for each side of the cross-section.
The rectangular faces use the length of the prism as one side.
If one length is in mm and another in cm, convert before you multiply.
The answer is in cm² or m².
A long thin box can have a large surface area and a small volume.
Do not multiply all three lengths and call that the surface area. That is the volume.
Sketch the solid, or list the faces in words, before you calculate. A cuboid has a top and a bottom, a front and a back, and two ends. That is three pairs. For edges 5 cm, 4 cm and 3 cm the pairs are 5 × 4 = 20, 5 × 3 = 15 and 4 × 3 = 12. Add the three areas to get 47, then double, because each rectangle has an opposite face: 2 × 47 = 94 cm^2. The bracket form 2(5 × 4 + 5 × 3 + 4 × 3) is the same arithmetic.
Missing a pair is the usual error. If you add 20 + 15 + 12 and stop, you have counted only half the solid. If you multiply 5 × 4 × 3 = 60, you have found the volume in cm^3, not the surface. The unit is the check. Surface area ends in cm^2. If your working never doubled a face and never added a second shape, look again at the list.
Method: 1. List the three pairs of faces, add one of each pair, then double. 2. Edges 5cm, 4cm and 3cm give pairs 20, 15 and 12. 3. The surface area is 2×(20+15+12)=94cm^2.
A cube is a cuboid whose three edges are equal. Every face is a square, and there are six of them. For side 5 cm, one face has area 5 × 5 = 25 cm^2. Six faces have area 6 × 25 = 150 cm^2. That is 6s^2, six copies of one face. It is not the volume, which would multiply the three edges as 5 × 5 × 5.
The volume of this cube is 125 cm^3. The surface area is 150 cm^2. The numbers are different, and the units are different. Surface area multiplies the area of one face by six. Volume multiplies the three edges together. On a question that asks for surface area, write cm^2 and show the six faces.
Method: 1. Find one square face, then multiply by 6. 2. Side 5cm gives one face of 5× 5=25cm^2. 3. Six faces give 6× 25=150cm^2.
A triangular prism has five faces. The two ends are the same triangle. The three sides are rectangles, and each rectangle uses one side of the triangle as its width and the length of the prism as its length. For a right-angled end with legs 6 cm and 8 cm, the third side is 10 cm. If the prism is 12 cm long, the rectangles are 6 × 12, 8 × 12 and 10 × 12.
The area of one right-angled end is 1/2 × 6 × 8 = 24 cm^2. Two ends make 48 cm^2. The rectangles make 72 + 96 + 120 = 288 cm^2. The total surface area is 48 + 288 = 336 cm^2. Leave the triangles in. A prism is not a tube: the ends are part of the outside unless the question says the solid is open.
Method: 1. Add two triangular ends to three rectangular sides. 2. Ends 1/2× 6× 8=24 make 48cm^2 together. 3. Rectangles 6× 12, 8× 12 and 10× 12 make 288cm^2, so the total is 336cm^2.
Write the unit on every area, and square it. Lengths in centimetres give areas in cm^2. If you mix millimetres and centimetres in the same sum, convert first so every length is in the same unit. A face of 8 cm by 5 cm is 40 cm^2. Doubling the three pairs of an 8 by 5 by 2 cuboid gives 2(40 + 16 + 10) = 132 cm^2.
A quick check is to see whether the answer is bigger than the largest face, and bigger than twice that face. For the 8 by 5 by 2 cuboid the largest face is 40 cm^2, and six faces of various sizes should land well above 80. 132 passes that check. An answer of 40 has counted one face. An answer of 80 has counted the volume, 8 × 5 × 2. Compare the question word “surface” with the unit you are about to write.
Method: 1. Convert every edge into centimetres, then give the answer in cm^2. 2. For an 8 by 5 by 2 cuboid, work out 2(40+16+10)=132cm^2. 3. Check the answer is larger than the biggest face, which is 40cm^2.
Every prism has two identical ends, called the cross-section, and a set of rectangular faces joining them. If you open the prism out, those rectangles form one long strip. The strip is as long as the perimeter of the cross-section and as wide as the length of the prism. So the surface area of any prism is 2 × area of the end + perimeter of the end × length.
This one rule works for triangular prisms, trapezium prisms and L-shaped prisms. Work out the cross-section area first, then the perimeter, then combine them. Be careful that the perimeter includes every side of the end, including the sloping ones, and that the "length" is the distance between the two ends, not the height of the picture.
Method: 1. Find the area of one end using the correct formula for that shape. 2. Double it to get both ends. 3. Add up every side of the end to get the perimeter, then multiply by the length of the prism. 4. Add the two parts and write the answer in square units.
Tip: Count the faces first. A triangular prism has 5 faces and a trapezium prism has 6. If your working has a different number of parts, something is missing.
Real exam questions often hide a length. You might be given the volume of a cube and asked for its surface area, so you must first find the edge: a cube of volume 64 cm^3 has edge 4 cm. A prism may give a slant height and a base, so you find a missing height with Pythagoras before you can use it. Do the extra step first and write it down.
Comparison questions give you a cuboid, colour some faces, and ask which colour covers more area. Work out the three different face areas, remember that each appears twice, and total each colour separately. Finish with a clear sentence. A common mistake is to compare single faces, when the question asks about the totals.
Method: 1. List the three different face areas l × w, l × h and w × h. 2. Double each one because every face has an opposite partner. 3. Add the faces that are in each group and compare the totals. 4. Write a sentence that answers the question asked, using both totals.
Surface area is used for real jobs: how much paint to cover a box, how much wrapping paper for a present, how many tiles for a tank. These questions add a final step. You compare the area you need with the area one tin or sheet covers, then round up to a whole number of tins, because you cannot buy part of a tin. Then multiply by the price.
Check the units before you compare. A paint tin may cover in square metres while your box is measured in centimetres. Convert one of them, remembering that 1 m^2 = 10 000 cm^2. If the question says two coats, double the area. Show each stage: area, converted area, number of tins, cost. Each stage is usually a separate mark.
Method: 1. Work out the surface area of the faces that will be covered. 2. Convert to the same unit as the coverage, using 1 m^2 = 10 000 cm^2. 3. Divide by the area one tin covers and round up to a whole tin. 4. Multiply the number of tins by the cost of one tin.
Tip: Always round up when buying. A value like 2.4 tins means you buy 3.
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