Matemáticas
Surface Area
Add up every face, and answer in cm² not cm³. The six faces of a cuboid totalled, then the curved part and the two ends of a cylinder worked in full.
Surface area
Surface area is the total area of all the faces and curved surfaces of a solid added together, measured in square units such as cm².
- También llamado
- Total surface area
- Dónde lo encuentra el alumnado
- Grade 6 with cuboids and prisms, extended to cylinders, cones and spheres by Grade 9, and on every GCSE and IGCSE mathematics specification.
La respuesta corta
Surface area is the total area of every face of a solid added together, measured in square units such as cm². Volume measures the space inside and uses cubic units. A cuboid measuring 8 cm by 5 cm by 3 cm has a surface area of 158 cm² and a volume of 120 cm³.
Un ejemplo
2(8 × 5) + 2(8 × 3) + 2(5 × 3) = 80 + 48 + 30 = 158 cm²
A cuboid has six rectangular faces in three matching pairs: front and back, top and bottom, left and right. Each pair uses two of the three dimensions, so working out three rectangles and doubling each is the whole method. The same solid has volume 8 × 5 × 3 = 120 cm³, which is a different quantity in different units and answers a different question.
The units tell you which one you found
Area is two-dimensional and its units are squared; volume is three-dimensional and its units are cubed. That is not a labelling convention added at the end, it is the arithmetic: multiplying two lengths gives cm × cm, and multiplying three gives cm × cm × cm.
This makes a free check available on every question. If a surface area answer has come out in cm³, three lengths have been multiplied together somewhere and volume has been found by mistake. Conversely a volume answer in cm² means a face has been calculated and the third dimension forgotten. Writing the units as you go, rather than adding them to the final line, catches the error while it is still one line old.
A cuboid is three pairs of rectangles
Every cuboid has six faces, and opposite faces are identical, so there are only three areas to calculate. With dimensions l, w and h the total is 2lw + 2lh + 2wh, but there is no need to memorise that as a formula if you can see the three pairs.
Sketching the net is the reliable way in for a student who keeps losing a face. Six rectangles laid flat can be counted, and counting is harder to get wrong than visualising. It also makes clear what changes when a face is missing — an open-topped box has five faces, and the question will say so.
A cylinder is a rectangle and two circles
Unroll the curved surface of a cylinder and it flattens into a rectangle. Its height is the height of the cylinder and its width is the distance round the circle, the circumference, so its area is 2πr × h. Add the two circular ends at πr² each and the total surface area is 2πrh + 2πr².
For a cylinder of radius 3 cm and height 10 cm, the curved surface is 2π × 3 × 10 = 60π cm² and the two ends come to 2π × 3² = 18π cm². The total is 78π, or 245 cm² to three significant figures. Leaving the answer as 78π is exact and often what a Higher paper wants.
Read the solid carefully before adding both ends. A pipe has no ends at all and a tin without a lid has one, so the two circles are not automatic. Questions that mention a label wrapped round a can, or a chimney, are testing exactly this.
Preguntas frecuentes
What is the difference between surface area and volume?
Surface area measures the outside of a solid — how much paper would wrap it — in square units. Volume measures the space inside — how much water would fill it — in cubic units. A question about painting, wrapping or covering wants surface area; a question about filling, capacity or how much material a solid contains wants volume.
Do I get the surface area formulae in the exam?
Some of them. Current GCSE papers give the sphere and cone formulae in the question or on a formulae sheet, but expect cuboids, prisms and cylinders to be worked out from first principles. Check the front of your board's paper rather than assuming, and learn the cylinder either way since it appears so often.
How do I find the surface area of a prism?
Two identical end faces plus the rectangles that wrap around them. The wrapping is a single rectangle whose width is the perimeter of the cross-section and whose length is the length of the prism, so surface area is 2 × (area of cross-section) + (perimeter of cross-section) × length. That works for triangular, trapezium and L-shaped prisms alike.
Why does doubling the dimensions not double the surface area?
Because area depends on two dimensions at once. Double every length of a cuboid and each face becomes four times as large, so the surface area quadruples while the volume goes up eight times. This is why large animals lose heat more slowly than small ones, and it is a standard Higher tier question in the form of area and volume scale factors.
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