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15.4 Calculating with prisms and cylinders

15.4 Calculating with prisms and cylinders

A prism is a 3D shape that has the same cross-section along its length.
Here are some examples of prisms. !e cross-section of each one is shaded.

Cross-section is a Cross-section is an Cross-section is Cross-section is
right-angled triangle. equilateral triangle. a trapezium. rectangle.

You can work out the volume of a prism using the formula: volume = area of cross-section × length

You can work out the surface area of a prism by $nding the
total area of all the faces of the prism.

Worked example 15.4a

a Work out the volume of each prism. i ii
b A prism has a volume of 91 cm . 5 cm 9 cm 6 cm
3
2
The area of the cross-section is 13 cm . 12 cm 2 8 cm
What is the length of the prism?
c Work out the surface area of this prism.
3 cm 7 cm
8 cm

a i V = area of cross-section × length The diagram shows the area of the cross-section of the cuboid.
= 12 × 5 Substitute the area and length into the formula for the volume.
3
3
= 60 cm Work out the answer and remember the units, cm .
1
ii Area of triangle = × base × height First, work out the area of the cross-section of the prism.
2
1
= × 8 × 9 Substitute base and height measurements in the area formula.
2
2
= 36 cm Work out the answer and remember the units, cm .
2
V = area of cross-section × length Now work out the volume of the prism, by substituting the area
= 36 × 6 and length into the volume formula.
3
= 216 cm Work out the answer and remember the units, cm .
3
b V = area of cross-section × length Write down the formula for the volume of a prism.
91 = 13 × l Substitute the volume and the area in the formula.
91
l = Rearrange the equation to make l the subject.
13
l = 7 cm Work out the answer and remember the units, cm.
1
c Area of triangle = × base × height First, work out the area of the triangular face.
2
1
= × 8 × 3 Substitute base and height measurements in the area formula.
2
= 12 cm Work out the answer.
2
2
Area of base = 8 × 7 = 56 cm The base is a rectangle, so work out the area (length × width).
Area of sloping side = 5 × 7 = 35 cm The side is a rectangle, so work out the area (length × width).
2
Surface area = 2 × 12 + 56 + 2 × 35 Now work out the total area. There are two triangular faces, one
= 150 cm base and two sloping sides. Remember the units, cm .
2
2
15 Area, perimeter and volume 147