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SAT Math : Solid Geometry

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Example Question #121 : Solid Geometry

A sphere fits inside of a cube so that its surface barely touches each side of the cube at any given time. If the volume of the box is 27 cubic centimeters, then what is the volume of the sphere?

Possible Answers:

$3\pi$

$\frac{27}{12}\pi$

$\frac{27}{6}\pi$

$6\pi$

None of the given answers.

Correct answer:

$\frac{27}{6}\pi$

Explanation:

If the volume of the cube is 27 cubic centimeters, then its height, width, and depth are all 3cm. Since the sphere fits perfectly in the cube, then the sphere's diameter is also 3. This means that its radius is $\frac{3}{2}$ .

Substitute this radius value into the equation for the volume of a sphere:

$V=\frac{4}{3}\pi r^3=\frac{4}{3}\cdot\pi \cdot(\frac{3}{2})^3=\frac{4}{3}\cdot\pi\cdot\frac{27}{8}$

$=\frac{108}{24}\pi = \frac{27}{6}\pi$

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Example Question #811 : Geometry

The surface area of a sphere is 100. Give its volume.

Possible Answers:

$\frac{500 \pi}{3 }$

$\frac{500 }{3 }$

$\frac{500 }{3 \pi}$

$\frac{500 \sqrt{\pi}}{3 }$

$\frac{500 \sqrt{\pi}}{3 \pi}$

Correct answer:

$\frac{500 \sqrt{\pi}}{3 \pi}$

Explanation:

The surface area of a sphere is equal to

$A= 4 \pi r ^{2}$ ,

where is the radius.

Setting A = 100 and solving for :

$4 \pi r ^{2} = 100$

$\frac{4 \pi r ^{2}}{4 \pi } = \frac{100}{4 \pi }$

$r ^{2} = \frac{25 }{\pi }$

$r = \sqrt{\frac{25 }{\pi } }$

Applying the Quotient of Radicals Rule:

$r = \frac{\sqrt{25 } }{\sqrt{\pi}}$

$r = \frac{5 }{\sqrt{\pi}}$

The volume of the sphere can be determined from the radius using the formula

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

so, setting $r = \frac{5 }{\sqrt{\pi}}$ :

$V = \frac{4}{3} \pi \cdot \left ( \frac{5 }{\sqrt{\pi}} \right ) ^{3}$

Applying the Quotient of Radicals Rule:

$V = \frac{4}{3} \pi \cdot \frac{5^{3} }{\left ( \sqrt{\pi}\right ) ^{3}}$

$= \frac{4}{3} \pi \cdot \frac{125 }{\pi \sqrt{\pi} }$

$= \frac{500}{3\sqrt{\pi}}$

Rationalizing the denominator:

$V = \frac{500 \cdot \sqrt{\pi}}{3\sqrt{\pi} \cdot \sqrt{\pi}} = \frac{500 \sqrt{\pi}}{3 \pi}$

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Example Question #41 : Spheres

is a point on a sphere, and is the point on the sphere farthest from . The shortest curve from to that is completely on the sphere is $24 \pi$ in length.

Give the volume of the sphere.

Possible Answers:

$36, 864 \pi$

$9,216 \pi$

$4,608 \pi$

$2,304 \pi$

$18,432 \pi$

Correct answer:

$18,432 \pi$

Explanation:

The figure referenced is below. is the center of the sphere.

Sphere

The shortest curve connecting to its opposite point is a semicircle. Also, $\overline{AB}$ is a diameter of the sphere, and $\overline{OA}$ and $\overline{OB}$ are radii.

Given radius , a semicircle has length

$l = \frac{1}{2} C= \frac{1}{2} \cdot 2 \pi r = \pi r$

Setting $l = 24\pi$ and solving for :

$\pi r = 24 \pi$

$\frac{\pi r }{\pi}= \frac{24 \pi}{\pi}$

r = 24 .

The volume of a sphere, given its radius , is

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

Setting r = 24 :

$V = \frac{4}{3} \pi \cdot 24 ^{3} = \frac{4}{3} \cdot 13,824 \cdot \pi = 18,432 \pi$

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Example Question #41 : Spheres

Find the volume of a sphere whose diameter is 10cm.

Possible Answers:

$100\pi$

$\frac{500}{3}\pi$

$\frac{400}{3}\pi$

$400\pi$

$\frac{25}{3}\pi$

Correct answer:

$\frac{500}{3}\pi$

Explanation:

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

Radius is half of the diameter. Half of 10cm is 5cm.

$Volume=\frac{4}{3}\pi 5^3$

$Volume=\frac{500}{3}\pi$

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Example Question #12 : How To Find The Volume Of A Sphere

Spheres element76

The radius of the sphere shown here is 5in. long. What is its volume?

Possible Answers:

$\frac{100\pi}{3}$ in.^3

$\frac{500\pi}{3} in.^3$

$100\pi$ in.^3

$\frac{125\pi}{3}in.^3$

$\frac{400\pi}{3}in.^3$

Correct answer:

$\frac{500\pi}{3} in.^3$

Explanation:

The formula for the volume of a sphere given its radius is

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

The radius is stated to be 5in. long; hence, we can calculate the sphere's volume by substituting this value for into the formula, as shown:

$V=\frac{4\pi (5)^3}{3}=\frac{4\pi(125)}{3}=\frac{500\pi}{3}$

Hence, the volume of the sphere is $\frac{500\pi}{3} in.^3$

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

The volume of a cylinder is 36π. If the cylinder’s height is 4, what is the cylinder’s diameter?

Possible Answers:

Correct answer:

Explanation:

Volume of a cylinder? V = πr²h. Rewritten as a diameter equation, this is:

V = π(d/2)²h = πd²h/4

Sub in h and V: 36p = πd²(4)/4 so 36p = πd²

Thus d = 6

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

A cylinder has a height of 5 inches and a radius of 3 inches. Find the lateral area of the cylinder.

Possible Answers:

30π

45π

24π

8π

15π

Correct answer:

30π

Explanation:

LA = 2π(r)(h) = 2π(3)(5) = 30π

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

A cylinder has a volume of 20. If the radius doubles, what is the new volume?

Possible Answers:

100

Correct answer:

Explanation:

The equation for the volume of the cylinder is πr²h. When the radius doubles (r becomes 2r) you get π(2r)²h = 4πr²h. So when the radius doubles, the volume quadruples, giving a new volume of 80.

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

A cylinder has a height that is three times as long as its radius. If the lateral surface area of the cylinder is 54π square units, then what is its volume in cubic units?

Possible Answers:

27π

81π

243π

54π

9π

Correct answer:

81π

Explanation:

Let us call r the radius and h the height of the cylinder. We are told that the height is three times the radius, which we can represent as h = 3r.

We are also told that the lateral surface area is equal to 54π. The lateral surface area is the surface area that does not include the bases. The formula for the lateral surface area is equal to the circumference of the cylinder times its height, or 2πrh. We set this equal to 54π,

2πrh = 54π

Now we substitute 3r in for h.

2πr(3r) = 54π

6πr² = 54π

Divide by 6π

r² = 9.

Take the square root.

r = 3.

h = 3r = 3(3) = 9.

Now that we have the radius and the height of the cylinder, we can find its volume, which is given by πr²h.

V = πr²h

V = π(3)²(9) = 81π

The answer is 81π.

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Example Question #821 : Sat Mathematics

What is the volume of a hollow cylinder whose inner radius is 2 cm and outer radius is 4 cm, with a height of 5 cm?

Possible Answers:

20π cm³

60π cm³

80π cm³

50π cm³

100π cm³

Correct answer:

60π cm³

Explanation:

The volume is found by subtracting the inner cylinder from the outer cylinder as given by V = πr_out² h – πr_in² h. The area of the cylinder using the outer radius is 80π cm³, and resulting hole is given by the volume from the inner radius, 20π cm³. The difference between the two gives the volume of the resulting hollow cylinder, 60π cm³.

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SAT Math : Solid Geometry

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #121 : Solid Geometry

Example Question #811 : Geometry

Example Question #41 : Spheres

Example Question #41 : Spheres

Example Question #12 : How To Find The Volume Of A Sphere

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

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

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

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

Example Question #821 : Sat Mathematics

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