Volume and surface area of solids

Advanced Math 7 · Unit 13 · volume-surface-3d · Teacher edition

volume-surface-3d

Name ______________________________Date ______________

1.
A cylinder has radius $3$ cm and height $10$ cm. Its volume is $k\pi$ cm³. Enter $k$ .
Answer: ______________
2.
A cone has radius $6$ cm and height $8$ cm. Its total surface area is $k\pi$ cm². Enter $k$ .
Answer: ______________
3.
A grain silo is a cylinder of radius $3$ m and height $10$ m, topped by a hemisphere of radius $3$ m. The volume of the silo is $k\pi$ m³. Enter $k$ .
Answer: ______________
4.
A square pyramid has a base edge of $6$ cm and a height of $5$ cm. What is its volume?
1. (A)
  $10$ cm³
2. (B)
  $60$ cm³
3. (C)
  $90$ cm³
4. (D)
  $180$ cm³
5.
A cylindrical jug of radius $4$ cm is filled with water to a height of $9$ cm. All the water is poured into an empty cone-shaped glass of radius $6$ cm, which it exactly fills. What is the height of the cone, in cm?
1. (A)
  $4$
2. (B)
  $6$
3. (C)
  $9$
4. (D)
  $12$
5. (E)
  $36$

Answer key — Volume and surface area of solids

1.
90volume-surface-3d-01
$V = \pi r^2 h = \pi \times 9 \times 10 = 90\pi$ , so $k = \mathbf{90}$ .
2.
96volume-surface-3d-02
$\ell = \sqrt{36 + 64} = 10$ . $S = 36\pi + 60\pi = 96\pi$ , so $k = \mathbf{96}$ . (Using $h = 8$ in place of $\ell$ gives $36\pi + 48\pi = 84\pi$ , the classic slip.)
3.
108volume-surface-3d-03
Cylinder $90\pi$ ; hemisphere $\frac{2}{3}\pi(27) = 18\pi$ . Total $108\pi$ , so $k = \mathbf{108}$ . (Using a full sphere gives $126\pi$ .)
4.
(B)
$60$ cm³
volume-surface-3d-04
$V = \frac{1}{3} \times 36 \times 5 = \mathbf{60}$ cm³. ( $90$ uses $\frac{1}{2}Bh$ — a pyramid is a third, not a half; $180$ is the full prism; $10$ uses the edge $6$ instead of the base area $36$ .)
5.
(D)
$12$
volume-surface-3d-05
$144\pi = \frac{1}{3}\pi(36)h = 12\pi h \Rightarrow h = \mathbf{12}$ cm. ( $4$ forgets the $\frac{1}{3}$ and solves $36h = 144$ ; $9$ assumes the height is unchanged.)