Year 9AC9M9M01

Volume and Surface Area of Prisms and Cylinders

Solve problems involving the volume and surface area of right prisms and cylinders using appropriate units.

One idea covers every prism and the cylinder: find the end face, then push it along.

Method

Volume of any prism

V = A_{\text{cross-section}} \times h

Area of the end, times the length.

Cylinder volume

V = \pi r^2 h

The end face is a circle.

Surface area is every face added

SA = 2 \times \text{ends} + \text{sides}

List the faces before calculating.

Cylinder surface area

SA = 2\pi r^2 + 2\pi r h

Two circles plus the curved side unrolled into a rectangle.

Units: squared for area, cubed for volume

e.g.

\text{A cylinder has } r = 4 \text{ cm and } h = 10 \text{ cm. Find its volume and surface area.}

Write the volume formula. $V = \pi r^2 h$
Substitute and keep $\pi$ exact. $V = \pi(4)^2(10) = 160\pi$
Round at the end. $V \approx 502{,}7 \text{ cm}^3$
For surface area, list the faces: two circles and one curved side. $2\pi r^2 + 2\pi r h$
Substitute. $2\pi(16) + 2\pi(4)(10) = 32\pi + 80\pi$
Add and round. $112\pi \approx 351{,}9 \text{ cm}^2$

Find the cross-section first. Then decide whether the question wants inside or outside.

\text{Prism, end area } 24, \text{ length } 9. \text{ Volume?}

Answer

216 \text{ units}^3

\text{Cylinder } r=3,\ h=7. \text{ Volume?}

Answer

63\pi \approx 197{,}9

\text{Cylinder } r=5,\ h=12. \text{ Curved surface only?}

Answer

2\pi(5)(12) = 120\pi \approx 377{,}0

\text{Triangular prism, end base } 6, \text{ height } 8, \text{ length } 15. \text{ Volume?}

Answer

360 \text{ units}^3

\text{Why does the curved side unroll into a rectangle?}

Answer

\text{Its width is the circumference, } 2\pi r.

\text{A cylinder's radius doubles. What happens to its volume?}

Answer

\text{It quadruples, since } r \text{ is squared.}