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Lesson overview

Geometry - surface area of 3D solids

Surface area is the total area of every face on the outside of a 3D solid -- imagine unfolding it flat into a net, then adding up the area of each piece. A cuboid has 3 pairs of matching rectangular faces. A prism has 2 matching cross-sections plus rectangles running around the sides. Surface area is measured in square units, even though the solid itself is 3D.

A cuboid has 3 pairs of matching faces

This cuboid has length 5 cm, depth 3 cm and height 4 cm.

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Top + bottom2 × (5×3) = 30

Front + back2 × (5×4) = 40

Left + right2 × (3×4) = 24

Each pair of opposite faces is identical, so find one of each pair and double it.

A prism's surface area includes both cross-sections

This right-angled triangular prism has legs 3 cm and 4 cm, and length 10 cm.

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The 3 rectangular side faces together have area equal to the cross-section's perimeter times the prism's length.

Surface area vs volume: different questions

For the same cuboid, surface area and volume use the same three numbers, but combine them differently.

Surface area $94 {cm}^{2}$

how much material covers the outside

Volume $60 {cm}^{3}$

how much space is inside

Surface area answers 'how much wrapping paper?'. Volume answers 'how much fits inside?'.

Worked example

Find the surface area of a cuboid measuring 6 cm by 4 cm by 3 cm.

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Find the area of each of the 3 different faces: 6×4=24, 6×3=18, 4×3=12.
Add these three areas: 24 + 18 + 12 = 54.
Double the total because each face has a matching opposite face: 54 × 2 = 108.
Write the final answer in square units: 108 ${cm}^{2}$ .

Try it

Find the area of each different face first, add them, then double the total for a cuboid.

Question 1

A cuboid measures 2 cm by 3 cm by 4 cm. What is its surface area?

Question 2

A cube has side length 5 cm. What is its surface area?

Question 3

A triangular prism has two cross-sections of area 8 ${cm}^{2}$ each, and 3 rectangular faces with total area 90 ${cm}^{2}$ . What is its total surface area?

Question 4

Which unit is correct for surface area?

Lengthcm

Surface area?

Volume ${cm}^{3}$

Question 5

A cuboid has faces of area 12 ${cm}^{2}$ , 20 ${cm}^{2}$ and 15 ${cm}^{2}$ . What is its total surface area?

Common mistakes

Watch for these when working through the lesson.

Forgetting to double the total, since every face has a matching opposite face.
Writing cubic units ( ${cm}^{3}$ ) instead of square units ( ${cm}^{2}$ ) for surface area.
For a prism, forgetting the two cross-section faces and only adding the rectangular sides.

Surface area of cuboids and prisms

Lesson overview

Worked example

Try it

Common mistakes

Related topics

Independent practice will plug in here