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High School Math : Spheres

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

The radius of a sphere is . What is the approximate volume of this sphere?

Possible Answers:

$20\pi$

$288\pi$

$516\pi$

$300\pi$

$138\pi$

Correct answer:

$288\pi$

Explanation:

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

$V=\frac{4}{3}\pi(6)^3$

$V=\frac{4}{3}\pi(216)$

$V=288\pi$

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

A cube has a side dimension of 4. A sphere has a radius of 3. What is the volume of the two combined, if the cube is balanced on top of the sphere?

Possible Answers:

$4 + 36\pi$

$16 + 36\pi$

$16 + 6\pi$

$64 + 36\pi$

$64 + 6\pi$

Correct answer:

$64 + 36\pi$

Explanation:

$V_{cube}=(\text{side})^3=(4)^3=64$

$V_{sphere}=\frac{4}{3}\pi r^3=\frac{4}{3}\pi(3^3)=\frac{4}{3}\pi(27)=36\pi$

$V_{total}=64+36\pi$

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Example Question #1 : Spheres

What is the volume of a sphere with a diameter of 6 in?

Possible Answers:

$288\pi\ in^{3}$

$36\pi\ in^{3}$

$216\pi\ in^{3}$

$108\pi\ in^{3}$

$72\pi\ in^{3}$

Correct answer:

$36\pi\ in^{3}$

Explanation:

The formula for the volume of a sphere is:

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

where = radius. The diameter is 6 in, so the radius will be 3 in.

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Example Question #181 : Psat Mathematics

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:

$\frac{4r}{3}$

$\frac{9r}{2}$

$\frac{9}{2r}$

$\frac{3}{r}$

$3r$

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 #3 : How To Find The Volume Of A Sphere

If the diameter of a sphere is , find the approximate volume of the sphere?

Possible Answers:

$95\pi$

$143\pi$

$205\pi$

$334\pi$

$167\pi$

Correct answer:

$167\pi$

Explanation:

The volume of a sphere = $\frac{4}{3}\pi r^{3}$

Radius is $\frac{1}{2}$ of the diameter so the radius = 5.

$V=\frac{4}{3}\pi5^{3}$

or $V=\frac{500}{3\pi}$

which is approximately $167\pi$

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Example Question #671 : Geometry

What is the surface area of a sphere with a radius of 15?

Possible Answers:

$665\pi$

$450\pi$

$900\pi$

$225\pi$

Correct answer:

$900\pi$

Explanation:

To solve for the surface area of a sphere you must use the equation

First, plug in 15 for and square it

$SA=4(15^2)\pi$

Multiply by 4 and to get

$900\pi=225(4)(\pi)$

The answer is $900\pi$ .

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

What is the surface area of a sphere whise radius is r=1.5cm .

Possible Answers:

$9\pi cm^{2}$

$\dpi{100} 18\pi cm^{2}$

Not enough information to solve

$\dpi{100} 4.5\pi cm^{2}$

$\dpi{100} 12.14 cm^{2}$

Correct answer:

$9\pi cm^{2}$

Explanation:

The surface area of a sphere is found by the formula $SA=4\pi r^{2}$ using the given radius of r=1.5cm .

$SA=4\pi (1.5cm)^{2}$

$SA=4*\pi*2.25cm^{2}$

$\rightarrow 9\pi cm^{2}$

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

Find the surface area of a sphere whose diameter is $D=\pi$ .

Possible Answers:

$\dpi{100} 2\pi^{2}$

37.7

$\dpi{100} 4\pi^{2}$

$\pi ^{3}$

$\dpi{100} 2\pi^{3}$

Correct answer:

$\pi ^{3}$

Explanation:

The surface area of a sphere is found by the formula $SA=4\pi r^{2}$ . We need to first convert the given diameter of $D=\pi$ to the sphere's radius.

D=2r

$\pi =2r$

$r=\pi/2$

Now, we can solve for surface area.

$\dpi{100} \dpi{100} SA=4\pi \left ( \frac{\pi }{2}\right)^{2}$

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

$\dpi{100} \rightarrow \pi^{3}$

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Example Question #4 : How To Find The Surface Area Of A Sphere

To the nearest tenth of a square centimeter, give the surface area of a sphere with volume 1,000 cubic centimeters.

Possible Answers:

$314.2 \textrm{ cm}^{2}$

$333.3 \textrm{ cm}^{2}$

$120.9 \textrm{ cm}^{2}$

$483.6 \textrm{ cm}^{2}$

$708.3 \textrm{ cm}^{2}$

Correct answer:

$483.6 \textrm{ cm}^{2}$

Explanation:

The volume of a sphere in terms of its radius is

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

Substitute V = 1,000 and solve for :

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

$\frac{4\pi r^3}{3} = 1,000$

$\frac{4\pi r^3}{3}\cdot \frac{3}{4\pi }= 1,000\cdot \frac{3}{4\pi }$

$r^3= \frac{3,000}{4\pi }$

$r=\sqrt[3]{ \frac{3,000}{4\pi }} \approx 6.20 \textrm{ cm }$

Substitute for in the formula for the surface area of a sphere:

$A = 4\pi r^2 \approx 4 \cdot \pi \cdot 6.20 ^2 \approx 483.6 \textrm{ cm}^{2}$

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

Find the surface area of a sphere with a radius of .

Possible Answers:

$42\Pi$

$4\Pi$

$16\Pi$

$36\Pi$

$64\Pi$

Correct answer:

$64\Pi$

Explanation:

The standard equation to find the area of a sphere is $4\Pi r^2$ .

Substitute the given radius into the standard equation to get the answer:

$4\Pi (4)^2=4\Pi (16)=64\Pi$

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High School Math : Spheres

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

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

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

Example Question #1 : Spheres

Example Question #181 : Psat Mathematics

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

Example Question #671 : Geometry

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

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

Example Question #4 : How To Find The Surface Area Of A Sphere

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

All High School Math Resources

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