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SAT Math : Solid Geometry

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Example Questions

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

Carlos has a pool in the shape of a rectangular prism. He fills the pool with a hose that ejects water at a rate of p gallons per minute. The bottom of the pool is A meters wide and B meters long. If there are k gallons in a cubic meter, then which of the following expressions will be equal to the amount of time it takes, in hours, for the water in the pool to reach a height of c meters?

Possible Answers:

ABcp/(60k)

60ABc/(pk)

ABck/(60p)

60ABcp/k

ABc/(pk)

Correct answer:

ABck/(60p)

Explanation:

Fillpool1

Fillpool2

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Example Question #771 : Sat Mathematics

A given cube with a volume of is split into two equal prisms. What is the volume of one of the prisms?

Possible Answers:

$\frac{30}{\sqrt{2}}$

$\frac{30}{7}$

$\frac{20}{\sqrt{5}}$

Correct answer:

Explanation:

The question tells us that the cube is split into two prisms of equal size. Therefore, we don't even need to use the formula for the volume of a prism. If the cube is split into two equal forms, then the two prisms will be equal to one-half of the cube's volume.

$40 \div 2 = 20$

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

A cube with volume 27 cubic inches is inscribed inside a sphere such that each vertex of the cube touches the sphere. What is the radius, in inches, of the sphere?

Possible Answers:

8.5

√3/2 (approximately 1.73)

(3√3)/2 (approximately 2.60)

Correct answer:

(3√3)/2 (approximately 2.60)

Explanation:

We know that the cube has a volume of 27 cubic inches, so each side of the cube must be ∛27=3 inches. Since the cube is inscribed inside the sphere, the diameter of the sphere is the diagonal length of the cube, so the radius of the sphere is half of the diagonal length of the cube. To find the diagonal length of the cube, we use the distance formula d=√(3²+3²+3² )=√(3*3² )=3√3, and then divide the result by 2 to find the radius of the sphere, (3√3)/2.

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

The surface area of a sphere is 100π square feet. What is the radius in feet?

Possible Answers:

100

Correct answer:

Explanation:

S = 4π(r²)

100π = 4π(r²)

100 = 4r²

25 = r²

5 = r

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Example Question #3 : How To Find The Radius Of A Sphere

What is the radius of a sphere with a surface area of 16?

Possible Answers:

$\frac{2}{\sqrt{2\pi}}$

$\frac{4}{\sqrt{\pi}}$

$\frac{2}{\sqrt{\pi}}$

Correct answer:

$\frac{2}{\sqrt{\pi}}$

Explanation:

In order to find the radius, first write the surface area formula for a sphere and substitute the surface area.

$A=4\pi r^2$

$16=4\pi r^2$

Divide by $4\pi$ on both sides in order to isolate the r^2 term.

$\frac{16}{4\pi}=\frac{4\pi r^2}{4\pi}$

Simplify both sides of the equation.

$r^2 = \frac{4}{\pi}$

Square root both sides.

$r= \sqrt {\frac{4}{\pi}} = \frac{\sqrt4}{\sqrt{\pi}} = \frac{2}{\sqrt{\pi}}$

The radius is $\frac{2}{\sqrt{\pi}}$ .

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

Find the diameter of a sphere with a surface area of $20\pi$ .

Possible Answers:

$2\sqrt5$

$\sqrt{10}$

$\sqrt5$

$2\sqrt{10}$

Correct answer:

$2\sqrt5$

Explanation:

Write the formula to find the surface area of a sphere.

$A=4\pi r^2$

Substitute the area and solve for the radius.

$20\pi=4\pi r^2$

r^2=5

$r=\sqrt5$

The diameter is double the radius.

$d=2\sqrt5$

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

What is the diameter of a sphere if the surface area is $64\pi$ ?

Possible Answers:

$4\sqrt3$

$8\sqrt3$

Correct answer:

Explanation:

Write the formula for the surface area of a sphere.

$A=4\pi r^2$

Substitute the area and find the radius.

$64\pi=4\pi r^2$

16=r^2

r=4

The diameter is double the radius.

d=8

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Example Question #781 : Geometry

What is the diameter of a sphere with a volume of ?

Possible Answers:

$\frac{3}{\pi ^3}$

$2\sqrt[3]{\frac{18}{\pi}}$

$2\sqrt[3]{\frac{9}{\pi}}$

$\sqrt[3]{\frac{9}{\pi}}$

$\sqrt[3]{\frac{18}{\pi}}$

Correct answer:

$2\sqrt[3]{\frac{9}{\pi}}$

Explanation:

Write the formula for the volume of a sphere.

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

Substitute the volume.

$12=\frac{4}{3}\pi r^3$

Multiply by $\frac{3}{4\pi}$ on both sides in order to isolate the r^3 term.

$12\cdot \frac{3}{4\pi}=\frac{4}{3}\pi r^3\cdot \frac{3}{4\pi}$

$\frac{9}{\pi}= r^3$

Cube root both sides.

$\sqrt[3]{\frac{9}{\pi}}= r$

The diameter is double the radius.

$d= 2\sqrt[3]{\frac{9}{\pi}}$

The answer is: $2\sqrt[3]{\frac{9}{\pi}}$

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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:

16π

128π

32π

256π

64π

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:

3/4

1/2

3/2

2/3

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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SAT Math : Solid Geometry

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

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

Example Question #771 : Sat Mathematics

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

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

Example Question #3 : How To Find The Radius Of A Sphere

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

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

Example Question #781 : Geometry

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

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

All SAT Math Resources

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