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Intermediate Geometry : Solid Geometry

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #1233 : Intermediate Geometry

A sphere with a radius of $\frac{6}{5}$ is cut out of a cube that has a side length of . What is the volume of the resulting figure?

Possible Answers:

111.09

102.38

114.41

117.76

Correct answer:

117.76

Explanation:

Since the cube is bigger, we will be subtracting the volume of the sphere from the volume of the cube.

Start by recalling how to find the volume of a sphere.

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

Plug in the given radius to find the volume.

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{6}{5})^3=\frac{288}{125}\pi$

Next, recall how to find the volume of a cube:

$\text{Volume of Cube}=(\text{edge})^3$

Plug in the given side length to find the volume of the cube.

$\text{Volume of Cube}=5^3=125$

Finally, subtract the volume of the sphere from the volume of the cube.

$\text{Volume of Figure}=125-\frac{288}{125}\pi=117.76$

Make sure to round to places after the decimal.

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Example Question #1234 : Intermediate Geometry

A sphere with a radius of $\frac{8}{3}$ is cut out of a cube that has a side length of . What is the volume of the resulting figure?

Possible Answers:

134.59

129.05

122.45

136.57

Correct answer:

136.57

Explanation:

Since the cube is bigger, we will be subtracting the volume of the sphere from the volume of the cube.

Start by recalling how to find the volume of a sphere.

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

Plug in the given radius to find the volume.

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{8}{3})^3=\frac{2048}{81}\pi$

Next, recall how to find the volume of a cube:

$\text{Volume of Cube}=(\text{edge})^3$

Plug in the given side length to find the volume of the cube.

$\text{Volume of Cube}=6^3=216$

Finally, subtract the volume of the sphere from the volume of the cube.

$\text{Volume of Figure}=216-\frac{2048}{81}\pi=136.57$

Make sure to round to places after the decimal.

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Example Question #1235 : Intermediate Geometry

A sphere with a radius of $\frac{9}{2}$ is cut out of a cube that has a side length of . What is the volume of the resulting figure?

Possible Answers:

144.17

130.30

156.89

195.56

Correct answer:

130.30

Explanation:

Since the cube is bigger, we will be subtracting the volume of the sphere from the volume of the cube.

Start by recalling how to find the volume of a sphere.

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

Plug in the given radius to find the volume.

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{9}{2})^3=\frac{243}{2}\pi$

Next, recall how to find the volume of a cube:

$\text{Volume of Cube}=(\text{edge})^3$

Plug in the given side length to find the volume of the cube.

$\text{Volume of Cube}=8^3=512$

Finally, subtract the volume of the sphere from the volume of the cube.

$\text{Volume of Figure}=512-\frac{243}{2}\pi=130.30$

Make sure to round to places after the decimal.

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Example Question #31 : Spheres

True or false: A sphere with radius 1 has volume $\frac{4}{3} \pi$ .

Possible Answers:

True

False

Correct answer:

True

Explanation:

Given radius , the volume of a sphere can be calculated according to the formula

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

Set r = 1 :

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

The statement is true.

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Example Question #32 : Spheres

What is the volume of a sphere with surface area 1,000 square centimeters?

Possible Answers:

$2,974 \; cm^{3}$

$3,187 \; cm^{3}$

$3,028 \; cm^{3}$

$3,523 \; cm^{3}$

$3,532 \; cm^{3}$

Correct answer:

$2,974 \; cm^{3}$

Explanation:

Use the surface area formula to find the radius, then use the volume formula to find the volume.

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

$4\pi r^{2} =1,000$

$r^{2} =\frac{1,000}{4\pi }=\frac{250}{\pi }$

$r =\sqrt{\frac{250}{\pi }}$

$V=\frac{4}{3} \pi r^{3} = \frac{4}{3} \pi \left (\sqrt{\frac{250}{\pi }} \right )^{3} \approx 2,974$

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

The diameter of a sphere is $8\:m$ . What is the sphere's surface area?

Possible Answers:

$201.6\:m^2$

$201.1\:m^2$

$202.7\:m^2$

$202.3\:m^2$

$201.0\:m^2$

Correct answer:

$201.1\:m^2$

Explanation:

The equation to find the surface area of a sphere is: $Area = 4 \cdot \pi \cdot r^2$

The only information required to solve for the area is the radius. This information is given to us in a sense. If the diameter of the sphere is $8\:m$ , the radius must be $4\:m$ . This is because the radius is equivalent to half of the diameter.

Now that we know the radius, we can solve for the area by substituting in $4\:m$ for the value of , the radius.

$Area = 4 \cdot \pi \cdot (4^2)$

$A = 4 \cdot \pi \cdot 16$

$A = 64 \cdot \pi$

A = 201.062 m^2

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

If the radius of a sphere is $1\:unit$ , what is the sphere's surface area?

Possible Answers:

$\frac{4}{3}\pi\:units$

$2\pi\:units$

$8\pi\:units$

$\frac{16}{9}\pi\:units$

$4\pi\:units$

Correct answer:

$4\pi\:units$

Explanation:

Write the formula for surface area of a sphere:

$A=4\pi r^2$

Plug in the value of radius and solve for the surface area:

$A=4\pi (1)^2 = 4\pi$

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

The diameter of a sphere is $18\:cm$ . What is its surface area?

Possible Answers:

$1017.9\:cm^2$

$972.0\:cm^2$

$1017.8\:cm^2$

$1017.4\:cm^2$

$1017.3\:cm^2$

Correct answer:

$1017.9\:cm^2$

Explanation:

The surface area of a sphere is given by the equation: $SA=4\pi r^2$

The only given information is the diameter: $18\:cm$ . In order to solve for the surface area, the only necessary information is the radius. The missing variable (radius) can be calculated through the given diameter because diameter is twice the length of the radius. That is: d= 2(r)

Using that information, radius can be solved for by
18=2(r)
$(\frac{18\:cm}{2})=r$
$9\:cm=r$

Given that the radius is $9\:cm$ , this value can be substituted in to solve for the final surface area:

$SA= 4 \pi (9\:cm)^2$

$SA= 4(81\:cm^2)\pi$

$SA= 324\pi\:cm^2$

$SA=1017.88\:cm^2$

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

Find the surface area of a sphere with a circumference of $\small \small 10\pi\,cm$ .

Possible Answers:

$\small \small 25\pi\,cm^2$

$\small 200\pi\,cm^2$

$\small \small 100\pi\,cm^2$

$\small \small 20\pi\,cm^2$

$\small \small 400\pi\,cm^2$

Correct answer:

$\small \small 100\pi\,cm^2$

Explanation:

The formula for the surface area of a sphere is simply

$\small A=4\pi r^2$

However, the problem is that we are given the circumference rather than the radius. Nonetheless, this isn't too big of an obstacle, as the formula for circumference in terms of radius is simply.

$\small C=2\pi r$

Substituting or value gives

$\small 10\pi=2\pi r$

Solving for the radius gives

$\small r=5$

We can then substitute this value into our formula for surface area.

$\small A=4\pi(5^2)$

$\small A=100\pi$

Therefore, our surface area is $\small 100\pi\,cm^2$

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

If the radius of a sphere is , what is the surface area of the sphere?

Possible Answers:

$900\pi$

$675\pi$

$13500\pi$

$300\pi$

$225\pi$

Correct answer:

$900\pi$

Explanation:

Write the equation for the surface area of a sphere.

$A=4\pi r^2$

Substitute the radius and find the area.

$A=4\pi (15)^2 = 4\pi(225)=900\pi$

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Intermediate Geometry : Solid Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #1233 : Intermediate Geometry

Example Question #1234 : Intermediate Geometry

Example Question #1235 : Intermediate Geometry

Example Question #31 : Spheres

Example Question #32 : 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 #3 : How To Find The Surface Area Of A Sphere

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

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

All Intermediate Geometry Resources

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