# Surface Area

## Calculate the surface areas of the given basic solid shapes using standard formulae.

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This is level 3; Find the surface area of a variety of prisms. The diagrams are not to scale except the diagram in the last question.

 1 Find the surface area of this triangular prism.Length: 5cm, Base: 12cm, Height: 8cm, Sloping edges: 10cm. ☐ cm2 ☐ ✓ ✗ 2 A pentagonal prism has a length of 8cm.The area of its pentagonal faces is 43cm2 and the sides of the pentagons are each 5cm.Find its surface area. ☐ cm2 ☐ ✓ ✗ 3 An octagonal prism has a length of 9.8cm.The area of its octagonal faces is 278.9cm2 and the sides of the octagons are each 7.6cm.Find its surface area correct to one decimal place. ☐ cm2 ☐ ✓ ✗ 4 Find the surface area of a triangular prism if the area of its cross section is 83cm2, its length is 43cm and the three sides of the triangular ends add up to 42cm. ☐ cm2 ☐ ✓ ✗ 5 The cross section of a semi-transparent prism is a right angled triangle as shown in the diagram. Calculate the surface area of the prism if AB = 17cm, BC = 12cm and CD = 16cm. ☐ cm2 ☐ ✓ ✗ 6       The frame for a large tent is in the shape of an isosceles triangular prism.The height of the tent is 3.5m, the width of the tent (at the triangular ends) is 3.3m, the sloping edges of the tent are each 3.86m an the length of the tent is 4.3m. Calculate the area of canvas needed for the tent (excluding the bottom of the prism as that is where the ground sheet goes). Give your answer in square metres to the nearest square metre. ☐ cm2 ☐ ✓ ✗ 7       Find the surface area of a trapezoidal prism based on the views drawn to scale below. Drag the ruler to measure the diagram, each length should be measured to the nearest centimetre.Hold down the shift key [or shift] while you drag to rotate the ruler. Zoom into this web page to see the ruler more clearly. ☐ cm2 ☐ ✓ ✗
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This is Surface Area level 3. You can also try:
Level 1 Level 2 Level 4 Level 5 Level 6 Level 7 Level 8 Level 9

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## Description of Levels

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Level 1 - Find the surface area of shapes made up of cubes.

Level 2 - Find the surface area of a variety of cuboids.

Level 3 - Find the surface area of a variety of prisms.

Level 4 - Find the surface area of a variety of cylinders.

Level 5 - Find the surface area of a variety of cones.

Level 6 - Find the surface area of a variety of pyramids.

Level 7 - Find the surface area of a variety of spheres.

Level 8 - Find the surface area of composite shapes.

Level 9 - Mixed, more challenging questions involving surface area.

Volume - Find the volume of basic solid shapes.

Surface Area = Volume - Can you find the ten cuboids that have numerically equal volumes and surface areas? A challenge in using technology.

Exam Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).

More on 3D Shapes including lesson Starters, visual aids, investigations and self-marking exercises.

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## Surface Area Formulae

Cube: $$6s^2$$ where $$s$$ is the length of one edge.

Cuboid: $$2(lw + lh + wh)$$ where $$l$$ is the length, $$w$$ is the width and $$h$$ is the height of the cuboid.

Cylinder: $$2\pi rh + 2\pi r^2$$ where $$h$$ is the height (or length) of the cylinder and $$r$$ is the radius of the circular end.

Cone: $$\pi r(r+l)$$ where $$l$$ is the distance from the apex to the rim of the circle (sloping height) of the cone and $$r$$ is the radius of the circular base.

Cone: $$\pi r(r+\sqrt{h^2+r^2})$$ where $$h$$ is the height of the cone and $$r$$ is the radius of the circular base.

Square based pyramid: $$s^2+2s\sqrt{\frac{s^2}{4}+h^2}$$ where $$h$$ is the height of the pyramid and $$s$$ is the length of a side of the square base.

Rectangular based pyramid: $$lw+l\sqrt{\frac{w^2}{4}+h^2}+w\sqrt{\frac{l^2}{4}+h^2}$$ where $$h$$ is the height of the pyramid, $$l$$ is the length of the base and $$w$$ is the width of the base.

Sphere: $$4\pi r^2$$ where $$r$$ is the radius of the sphere.

Prism: Double the area of the cross section added to the product of the length and the perimeter of the cross section.

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