In this section we calculate the volume and also surface area the 3-D shapes such together cubes, cuboids, prisms and also cylinders.

You are watching: Surface area of three dimensional figures

Cube
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Volume = x³Surface area = 6x²
Cuboid
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Volume = xyzSurface area = 2xy + 2xz + 2yz
Cylinder
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Volume = π r²hArea of curved surface = 2π rhArea that each finish = π r²Total surface ar area = 2π rh + 2π r²
Prism
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A prism has a uniform cross-sectionVolume = area of cross ar × size = A l

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Calculate the volume the the cuboid shown.


Calculate the surface area that the cuboid shown.


Example 2

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Calculate the volume and also total surface area of the cylinder shown.



Volume = π r²h = π × 4² × 6 = 96 π
= 301.5928947 cm³
= 302 cm³ (to 3 far-ranging figures)
Area of curved surface = 2π rh = 2 × π × 4 × 6
= 48π
= 150.7964474 cm²
Area of each end = π r² = π × 4²
= 16π
= 50.26548246 cm²
Total surface ar area = 150.7964474 + (2 × 50.26548246)
= 251.3274123 cm²
= 251 cm² (to 3 significant figures)

Note: native the functioning we deserve to see that the area of the curved surface ar is 48π, and also that the area of each finish is 16π. The total surface area is therefore

48π + (2 × 16π) = 80π = 251.3274123 cm²
= 251 cm² (to 3 far-reaching figures)

concern 2

Giving her answers exactly to 3 far-ranging figures, calculation the volume and total surface area of every of the following cylinders:


inquiry 5
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The diagram shows a wood block that has had a hole drilled in it. The diameter of the feet is 2 cm.Calculate the volume that this solid, giving your answer correct to 2 decimal places.


Volume = block – hole = 4 × 6 × 6 – 1² × π × 6 = 144 – 6π = 125.15 (to 2 d.p.)
inquiry 6
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A concrete beam is to remainder on two concrete pillars. The beam is a cuboid v sides of length 0.5m, 3m and 0.4m.The pillars have diameter 0.4m and also height 2m.Calculate the total volume the concrete needed to do the beam and the pillars. Round her answer to a judicious level that accuracy.


question 7

The diagram reflects the cross-section of a pipeline of length 50 cm.The inside diameter that the pipeline is 20 cm and also the external diameter is 30 cm.


Calculate the volume of metal needed to do the pipe. Round your answer to a judicious level of accuracy.

cm³

Calculate the total surface area the the pipe, including the within surface.Round your answer come 3 significant figures.

cm²
Total surface area = 2 × (15² – 10²) × π + 30π × 50 + 20π × 50
= 250π + 1500π + 1000π = 2750π
= 8639.379797 cm² = 8640 cm² (to 3 s.f.)

concern 8
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The diagram reflects a prism.The cross-section of the prism is composed of a rectangle and a semicircle.


concern 10

A cylinder has a diameter that 12 cm and a curved surface ar area of 132π or 415 cm² (to 3 far-ranging figures).

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These cuboids room made from small cubes. Write exactly how many little cubes there space in every cuboid. The first is done for you.

(i)
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12 tiny cubes
(ii)
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little cubes
(iii)
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tiny cubes
(iv)
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tiny cubes

Area the cross-section = × basic × height = × 6 × 8 = 24 cm²