Solid Geometry

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ACT Math › Solid Geometry

Questions 1 - 10

1

Find the surface area of a cube with side length .

Explanation

To solve, simply multiply the face area by . Thus,

2

If the surface area of a cube equals 96, what is the length of one side of the cube?

4

3

6

5

Explanation

The surface area of a cube = 6a2 where a is the length of the side of each edge of the cube. Put another way, since all sides of a cube are equal, a is just the lenght of one side of a cube.

We have 96 = 6a2 → a2 = 16, so that's the area of one face of the cube.

Solving we get √16, so a = 4

3

Find the volume of a cube with side length 10.

$10\sqrt{2}$

Explanation

To find volume, simply cube the side length. Thus,

4

Find the volume of a cube given side length is .

Explanation

To solve, simply cube the side length. Thus,

5

The slant height of a cone is ; the diameter of its base is one-fifth its slant height. Give the surface area of the cone in terms of .

$\frac{11\pi L^{2}}{100}$

$\frac{6\pi L^{2}}{25}$

$\frac{ \pi L^{2}}{10}$

$\frac{ \pi L^{2}}{5}$

$\frac{ \pi L^{2}}{20}$

Explanation

The formula for the surface area of a cone with base of radius and slant height is

$A = \pi r^{2} + \pi r L$ .

The diameter of the base is $\frac{1}{5}L$ ; the radius is half this, so

$r = \frac{1}{2} \cdot \frac{1}{5}L= \frac{1}{10}L$

Substitute in the surface area formula:

$A = \pi \left ( \frac{1}{10}L \right ) ^{2} + \pi \left ( \frac{1}{10}L \right ) L$

$A = \frac{1}{100} \pi L^{2} + \frac{1}{10} \pi L^{2}$

$A = \frac{1}{100} \pi L^{2} + \frac{10}{100} \pi L^{2}$

$A = \frac{11}{100} \pi L^{2} = \frac{11\pi L^{2}}{100}$

6

Find the volume of a cube with side length 10.

$10\sqrt{2}$

Explanation

To find volume, simply cube the side length. Thus,

7

Find the surface area of a cube whose side length is .

Explanation

To find surface area of a cube, simply calculate the area of one side and multiply it by . Thus,

8

Find the surface area of a cube with side length .

Explanation

To solve, simply multiply the face area by . Thus,

9

Find the surface area of a sphere whose radius is .

$36\pi$

$12\pi$

$108\pi$

$4\pi$

Explanation

To solve, simply remember the following formula:

$SA=4\pi{r^2}=4*\pi*3^2=36\pi$

10

Given the volume of a cube is , find the side length.

Explanation

To find side length, simply realize that volume is the side length cubed. Thus,

$s=\sqrt[3]{V}=\sqrt[3]{64}=\sqrt[3]{4^3}=4$

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