Volume and Surface Area: Additional Practice (Set 1)

Use $V = \pi r^2 h$. Square the radius first.

$r^2 = 6^2 = 36$

$V = 3.14 \times 36 \times 5 = 565.2 \text{ cm}^3$

Find one face of each kind, then double the total.

$lw = 8 \times 3 = 24, \quad lh = 8 \times 3 = 24, \quad wh = 3 \times 3 = 9$

$S = 2(24 + 24 + 9) = 2 \times 57 = 114 \text{ cm}^2$

Find the triangular base area first.

$B = \frac{1}{2} \times 8 \times 3 = 12 \text{ m}^2$

Then multiply by the length.

$V = 12 \times 6 = 72 \text{ m}^3$

Volume is base area times height, so divide.

$h = \frac{V}{\text{base area}} = \frac{350}{50} = 7 \text{ cm}$

The big cube has edge $6$ cm, so cube it.

$V = 6^3 = 216 \text{ cm}^3$

Check against the pieces: each small cube is $3^3 = 27 \text{ cm}^3$, and $8 \times 27 = 216 \text{ cm}^3$, the same total.

Divide the volume by the base area.

$\text{base area} = 7 \times 4 = 28 \text{ cm}^2$

$h = \frac{84}{28} = 3 \text{ cm}$

Read the dimensions $7$ cm, $5$ cm, and $6$ cm and multiply.

$V = 7 \times 5 \times 6 = 210 \text{ cm}^3$

Add both circular ends and the rolled-out side.

$2\pi r^2 = 2 \times 3.14 \times 25 = 157 \text{ cm}^2$

$2\pi r h = 2 \times 3.14 \times 5 \times 10 = 314 \text{ cm}^2$

$S = 157 + 314 = 471 \text{ cm}^2$

The base area is $s^2$ and volume is base area times height.

$s^2 \times 5 = 80 \;\Rightarrow\; s^2 = 16 \;\Rightarrow\; s = 4 \text{ cm}$

Each edge is $7$ cm, and a cube has $6$ square faces of area $s^2$.

$S = 6s^2 = 6 \times 7^2 = 6 \times 49 = 294 \text{ cm}^2$

Recasting keeps the volume the same, so just find the cylinder's volume.

$V = \pi r^2 h = 3.14 \times 2^2 \times 9 = 3.14 \times 4 \times 9 = 113.04 \text{ cm}^3$

The cube has the same $113.04 \text{ cm}^3$ of material.

Both volumes are base area times height, and both the base area and the height match.

$V = (\text{base area}) \times h$

Same base area and same height give the same volume, whatever the base shape. The shape of the base does not matter once its area is fixed.