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PSAT Math : Geometry

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Example Questions

Example Question #1 : Spheres

The surface area of a sphere is 676π in². How many cubic inches is the volume of the same sphere?

Possible Answers:

2028π

(2028π)/3

(8788π)/3

(2197π)/3

8788π

Correct answer:

(8788π)/3

Explanation:

To begin, we must solve for the radius of our sphere. To do this, recall the equation for the surface area of a sphere: A = 4πr²

For our data, that is: 676π = 4πr²; 169 = r²; r = 13

From this, it is easy to solve for the volume of the sphere. Recall the equation:

V = (4/3)πr³

For our data, this is: V = (4/3)π * 13³ = (4π * 2197)/3 = (8788π)/3

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Example Question #8 : How To Find The Volume Of A Sphere

What is the difference between the volume and surface area of a sphere with a radius of 6?

Possible Answers:

216π

144π

720π

288π

133π

Correct answer:

144π

Explanation:

Surface Area = 4πr² = 4 * π * 6² = 144π

Volume = 4πr³/3 = 4 * π * 6³ / 3 = 288π

Volume – Surface Area = 288π – 144π = 144π

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Example Question #1 : How To Find The Volume Of A Sphere

A sphere is perfectly contained within a cube that has a volume of 216 units. What is the volume of the sphere?

Possible Answers:

$48\pi$

$12\pi$

$9\pi$

$36\pi$

$288\pi$

Correct answer:

$36\pi$

Explanation:

To begin, we must determine the dimensions of the cube. This is done by solving the simple equation: V=s^3

We know the volume is 216, allowing us to solve for the length of a side of the cube.

216=s^3

Taking the cube root of both sides, we get s = 6.

The diameter of the sphere will be equal to side of the cube, since the question states that the sphere is perfectly contained. The diameter of the sphere will be 6, and the radius will be 3.

We can plug this into the equation for volume of a sphere:

$V=\frac{4}{3}\pi r^3$

$V=\frac{4}{3}\pi (3)^3$

We can cancel out the 3 in the denominator.

$V=4\pi (3)^2$

Simplify.

$V=4\pi (9)$

$V=36\pi$

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Example Question #2 : How To Find The Volume Of A Sphere

The surface area of a sphere is $36\pi\ mm^2$ . Find the volume of the sphere in cubic millimeters.

Possible Answers:

$6\pi\ mm^3$

$3\pi\ mm^3$

$9\pi\ mm^3$

$36\pi\ mm^3$

$4\pi\ mm^3$

Correct answer:

$36\pi\ mm^3$

Explanation:

$SA=4\pi r^2$

$36\pi=4\pi r^2$

36=4r^2

9=r^2

3=r

$V=\frac{4}{3}\pi r^3$

$V=\frac{4}{3}\pi(3)^3$

$V=\frac{4}{3}(27)\pi$

$V=4(9)\pi$

$V=36\pi$

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Example Question #13 : Spheres

A solid hemisphere has a radius of length r. Let S be the number of square units, in terms of r, of the hemisphere's surface area. Let V be the number of cubic units, in terms of r, of the hemisphere's volume. What is the ratio of S to V?

Possible Answers:

$\frac{3}{r}$

$\frac{9}{2r}$

$\frac{9r}{2}$

$\frac{4r}{3}$

$3r$

Correct answer:

$\frac{9}{2r}$

Explanation:

First, let's find the surface area of the hemisphere. Because the hemisphere is basically a full sphere cut in half, we need to find half of the surface area of a full sphere. However, because the hemisphere also has a circular base, we must then add the area of the base.

$S = \frac{1}{2}\cdot$ (surface area of sphere) + (surface area of base)

The surface area of a sphere with radius r is equal to $4\pi r^2$ . The surface area of the base is just equal to the surface area of a circle, which is $\pi r^2$ .

$S=\frac{1}{2}\cdot 4\pi r^2+\pi r^2=2\pi r^2+\pi r^2=3\pi r^2$

The volume of the hemisphere is going to be half of the volume of an entire sphere. The volume for a full sphere is $\frac{4}{3}\pi r^3$ .

$V = \frac{1}{2}\cdot$ (volume of sphere)

$V = \frac{1}{2}\cdot \frac{4}{3}\pi r^3=\frac{2}{3}\pi r^3$

Ultimately, the question asks us to find the ratio of S to V. To do this, we can write S to V as a fraction.

$\frac{S}{V}=\frac{3\pi r^2}{\frac{2}{3}\pi r^3}$

In order to simplify this, let's multiply the numerator and denominator both by 3.

$\frac{S}{V}=\frac{3\pi r^2}{\frac{2}{3}\pi r^3}$ = $\frac{9\pi r^2}{2\pi r^3}=\frac{9}{2r}$

The answer is $\frac{9}{2r}$ .

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Example Question #12 : Spheres

A water tank takes the shape of a sphere whose exterior has radius 16 feet; the tank is six inches thick throughout. To the nearest hundred, how many cubic feet of water does the tank hold?

Possible Answers:

$15,600\textrm{ ft}^{3}$

$2,800\textrm{ ft}^{3}$

$17,100\textrm{ ft}^{3}$

$3,100\textrm{ ft}^{3}$

$14,100\textrm{ ft}^{3}$

Correct answer:

$15,600\textrm{ ft}^{3}$

Explanation:

Six inches is equal to 0.5 feet, so the radius of the interior of the tank is

16-0.5 = 15.5 feet.

The amount of water the tank holds is the volume of the interior of the tank, which is

$V = \frac{4}{3} \pi r^{3}$

$V \approx \frac{4}{3} \cdot 3.14 \cdot 15.5^{3} \approx 15,591$ ,

which rounds to 15,600 cubic feet.

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Example Question #1 : How To Find The Surface Area Of A Sphere

A spherical orange fits snugly inside a small cubical box such that each of the six walls of the box just barely touches the surface of the orange. If the volume of the box is 64 cubic inches, what is the surface area of the orange in square inches?

Possible Answers:

64π

256π

128π

16π

32π

Correct answer:

16π

Explanation:

The volume of a cube is found by V = s³. Since V = 64, s = 4. The side of the cube is the same as the diameter of the sphere. Since d = 4, r = 2. The surface area of a sphere is found by SA = 4π(r²) = 4π(2²) = 16π.

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Example Question #1 : How To Find The Surface Area Of A Sphere

A solid sphere is cut in half to form two solid hemispheres. What is the ratio of the surface area of one of the hemispheres to the surface area of the entire sphere before it was cut?

Possible Answers:

1/2

3/4

2/3

3/2

Correct answer:

3/4

Explanation:

The surface area of the sphere before it was cut is equal to the following:

surface area of solid sphere = 4πr², where r is the length of the radius.

Each hemisphere will have the following shape:

In order to determine the surface area of the hemisphere, we must find the surface area of the flat region and the curved region. The flat region will have a surface area equal to the area of a circle with radius r.

area of flat part of hemisphere = πr²

The surface area of the curved portion of the hemisphere will equal one-half of the surface area of the uncut sphere, which we established to be 4πr².

area of curved part of hemisphere = (1/2)4πr²= 2πr²

The total surface area of the hemisphere will be equal to the sum of the surface areas of the flat part and curved part of the hemisphere.

total surface area of hemisphere = πr²+ 2πr²= 3πr²

Finally, we must find the ratio of the surface area of the hemisphere to the surface area of the uncut sphere.

ratio = (3πr²)/(4πr²) = 3/4

The answer is 3/4.

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Example Question #2 : Spheres

The volume of a sphere is 2304π in³. What is the surface area of this sphere in square inches?

Possible Answers:

None of the other answers

144π

576π

216π

36π

Correct answer:

576π

Explanation:

To solve this, we must first begin by finding the radius of the sphere. To do this, recall that the volume of a sphere is:

V = (4/3)πr³

For our data, we can say:

2304π = (4/3)πr³; 2304 = (4/3)r³; 6912 = 4r³; 1728 = r³; 12 * 12 * 12 = r³; r = 12

Now, based on this, we can ascertain the surface area using the equation:

A = 4πr²

For our data, this is:

A = 4π*12² = 576π

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Example Question #5 : Spheres

A sphere has its center at the origin. A point on its surface is found on the x-y axis at (6,8). In square units, what is the surface area of this sphere?

Possible Answers:

200π

(400/3)π

400π

None of the other answers

40π

Correct answer:

400π

Explanation:

To find the surface area, we must first find the radius. Based on our description, this passes from (0,0) to (6,8). This can be found using the distance formula:

6² + 8² = r²; r² = 36 + 64; r² = 100; r = 10

It should be noted that you could have quickly figured this out by seeing that (6,8) is the hypotenuse of a 6-8-10 triangle (which is a multiple of the "easy" 3-4-5).

The rest is easy. The surface area of the sphere is defined by:

A = 4πr² = 4 * 100 * π = 400π

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PSAT Math : Geometry

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : Spheres

Example Question #8 : How To Find The Volume Of A Sphere

Example Question #1 : How To Find The Volume Of A Sphere

Example Question #2 : How To Find The Volume Of A Sphere

Example Question #13 : Spheres

Example Question #12 : Spheres

Example Question #1 : How To Find The Surface Area Of A Sphere

Example Question #1 : How To Find The Surface Area Of A Sphere

Example Question #2 : Spheres

Example Question #5 : Spheres

All PSAT Math Resources

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