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ISEE Upper Level Math : How to find the surface area of a sphere

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Example Questions

Example Question #82 : Solid Geometry

In terms of $\pi$ $\displaystyle \pi$ , give the surface area, in square feet, of a spherical tank with diameter 36 inches.

Possible Answers:

$3 \pi \textrm{ ft}^{2}$ $\displaystyle 3 \pi \textrm{ ft}^{2}$

$36\pi \textrm{ ft}^{2}$ $\displaystyle 36\pi \textrm{ ft}^{2}$

$\frac{9}{2} \pi \textrm{ ft}^{2}$ $\displaystyle \frac{9}{2} \pi \textrm{ ft}^{2}$

$18 \pi \textrm{ ft}^{2}$ $\displaystyle 18 \pi \textrm{ ft}^{2}$

$9 \pi \textrm{ ft}^{2}$ $\displaystyle 9 \pi \textrm{ ft}^{2}$

Correct answer:

$9 \pi \textrm{ ft}^{2}$ $\displaystyle 9 \pi \textrm{ ft}^{2}$

Explanation:

36 inches = $36 \div 12 = 3$ $\displaystyle 36 \div 12 = 3$ feet, the diameter of the tank. Half of this, or $\frac{3}{2}$ $\displaystyle \frac{3}{2}$ feet, is the radius. Set $r = \frac{3}{2}$ $\displaystyle r = \frac{3}{2}$ , substitute in the surface area formula, and solve for $\displaystyle A$ :

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

$A =4\pi \cdot \left( \frac{3}{2} \right )^{2}$ $\displaystyle A =4\pi \cdot \left( \frac{3}{2} \right )^{2}$

$A =\frac{ 4}{1} \cdot \frac{3}{2}\cdot \frac{3}{2} \cdot \pi$ $\displaystyle A =\frac{ 4}{1} \cdot \frac{3}{2}\cdot \frac{3}{2} \cdot \pi$

$A =\frac{ 1}{1} \cdot \frac{3}{1}\cdot \frac{3}{1} \cdot \pi$ $\displaystyle A =\frac{ 1}{1} \cdot \frac{3}{1}\cdot \frac{3}{1} \cdot \pi$

$A =9 \pi$ $\displaystyle A =9 \pi$

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Example Question #402 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Give the surface area of a sphere with diameter $\displaystyle 8$ .

Possible Answers:

$64 \pi$ $\displaystyle 64 \pi$

$128 \pi$ $\displaystyle 128 \pi$

$32 \pi$ $\displaystyle 32 \pi$

$256 \pi$ $\displaystyle 256 \pi$

$\frac{256}{3} \pi$ $\displaystyle \frac{256}{3} \pi$

Correct answer:

$64 \pi$ $\displaystyle 64 \pi$

Explanation:

A sphere with diameter $\displaystyle 8$ has radius half that, or $\displaystyle 4$ , so substitute r = 4 $\displaystyle r = 4$ into the formula for the surface area of a sphere:

$A = 4 \pi r^{2} = 4 \pi \cdot 4^{2}= 64 \pi$ $\displaystyle A = 4 \pi r^{2} = 4 \pi \cdot 4^{2}= 64 \pi$

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Example Question #83 : Solid Geometry

A spherical buoy has a radius of 5 meters. What is the surface area of the buoy?

Possible Answers:

$100\pi m^2$ $\displaystyle 100\pi m^2$

$50\pi m^2$ $\displaystyle 50\pi m^2$

100 m^2 $\displaystyle 100 m^2$

50 m^2 $\displaystyle 50 m^2$

Correct answer:

$100\pi m^2$ $\displaystyle 100\pi m^2$

Explanation:

A spherical buoy has a radius of 5 meters. What is the surface area of the buoy?

To find the surface area of a sphere, use the following:

$SA_{sphere}=4\pi r^2$ $\displaystyle SA_{sphere}=4\pi r^2$

Plug in our radius and solve!

$SA_{sphere}=4\pi (5m)^2=100\pi m^2$ $\displaystyle SA_{sphere}=4\pi (5m)^2=100\pi m^2$

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Example Question #84 : Solid Geometry

You have a wooden ball which you are going to paint. If the radius is 12 inches, what is the surface area of the ball?

Possible Answers:

$576\pi in^2$ $\displaystyle 576\pi in^2$

$192\pi in^2$ $\displaystyle 192\pi in^2$

Not enough information provided

$144 \pi in^2$ $\displaystyle 144 \pi in^2$

Correct answer:

$576\pi in^2$ $\displaystyle 576\pi in^2$

Explanation:

You have a wooden ball which you are going to paint. If the radius is 12 inches, what is the surface area of the ball?

First, recall the formula for surface area of a sphere:

$SA_{sphere}=4 \pi r^2$ $\displaystyle SA_{sphere}=4 \pi r^2$

Now, just plug in our known radius and simplify:

$SA_{sphere}=4 \pi (12in)^2=576\pi in^2$ $\displaystyle SA_{sphere}=4 \pi (12in)^2=576\pi in^2$

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Example Question #5 : How To Find The Surface Area Of A Sphere

Find the surface area of a sphere with a diameter of 10in.

Possible Answers:

$100\pi \text{ cm}^2$ $\displaystyle 100\pi \text{ cm}^2$

$75\pi \text{ cm}^2$ $\displaystyle 75\pi \text{ cm}^2$

$36\pi \text{ cm}^2$ $\displaystyle 36\pi \text{ cm}^2$

$50\pi \text{ cm}^2$ $\displaystyle 50\pi \text{ cm}^2$

$125\pi \text{ cm}^2$ $\displaystyle 125\pi \text{ cm}^2$

Correct answer:

$100\pi \text{ cm}^2$ $\displaystyle 100\pi \text{ cm}^2$

Explanation:

To find the surface area of a sphere, we will use the following formula:

$SA = 4 \cdot \pi \cdot r^2$ $\displaystyle SA = 4 \cdot \pi \cdot r^2$

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 10cm. We also know the diameter is two times the radius. Therefore, the radius is 5cm.

Knowing this, we can substitute into the formula. We get

$SA = 4 \cdot \pi \cdot (5\text{cm})^2$ $\displaystyle SA = 4 \cdot \pi \cdot (5\text{cm})^2$

$SA = 4 \cdot \pi \cdot 25\text{cm}^2$ $\displaystyle SA = 4 \cdot \pi \cdot 25\text{cm}^2$

$SA = 100\text{cm}^2 \cdot \pi$ $\displaystyle SA = 100\text{cm}^2 \cdot \pi$

$SA = 100\pi \text{ cm}^2$ $\displaystyle SA = 100\pi \text{ cm}^2$

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Example Question #6 : How To Find The Surface Area Of A Sphere

Find the surface area of a sphere with a radius of 10in.

Possible Answers:

$375\pi \text{ in}^2$ $\displaystyle 375\pi \text{ in}^2$

$250\pi \text{ in}^2$ $\displaystyle 250\pi \text{ in}^2$

$125\pi \text{ in}^2$ $\displaystyle 125\pi \text{ in}^2$

$100\pi \text{ in}^2$ $\displaystyle 100\pi \text{ in}^2$

$400\pi \text{ in}^2$ $\displaystyle 400\pi \text{ in}^2$

Correct answer:

$400\pi \text{ in}^2$ $\displaystyle 400\pi \text{ in}^2$

Explanation:

To find the surface area of a sphere, we will use the following formula:

$SA = 4 \cdot \pi \cdot r^2$ $\displaystyle SA = 4 \cdot \pi \cdot r^2$

where r is the radius of the sphere.

Now, we know the radius of the sphere is 10in.

Knowing this, we can substitute into the formula. We get

$SA = 4 \cdot \pi \cdot (10\text{in})^2$ $\displaystyle SA = 4 \cdot \pi \cdot (10\text{in})^2$

$SA = 4 \cdot \pi \cdot 100\text{in}^2$ $\displaystyle SA = 4 \cdot \pi \cdot 100\text{in}^2$

$SA = 400\pi \text{ in}^2$ $\displaystyle SA = 400\pi \text{ in}^2$

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Example Question #13 : Spheres

Find the surface area of a sphere with a diameter of 18in.

Possible Answers:

$324\pi \text{ in}^2$ $\displaystyle 324\pi \text{ in}^2$

$421\pi \text{ in}^2$ $\displaystyle 421\pi \text{ in}^2$

$256\pi \text{ in}^2$ $\displaystyle 256\pi \text{ in}^2$

$212\pi \text{ in}^2$ $\displaystyle 212\pi \text{ in}^2$

$138\pi \text{ in}^2$ $\displaystyle 138\pi \text{ in}^2$

Correct answer:

$324\pi \text{ in}^2$ $\displaystyle 324\pi \text{ in}^2$

Explanation:

To find the surface area of a sphere, we will use the following formula:

$SA = 4 \cdot \pi \cdot r^2$ $\displaystyle SA = 4 \cdot \pi \cdot r^2$

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 18in. We also know the diameter is two times the radius. Therefore, the radius is 9in.

Knowing this, we can substitute into the formula. We get

$SA = 4 \cdot \pi \cdot (9\text{in})^2$ $\displaystyle SA = 4 \cdot \pi \cdot (9\text{in})^2$

$SA = 4 \cdot \pi \cdot 81\text{in}^2$ $\displaystyle SA = 4 \cdot \pi \cdot 81\text{in}^2$

$SA = 324 \pi \text{ in}^2$ $\displaystyle SA = 324 \pi \text{ in}^2$

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

Find the surface area of a sphere with a radius of 6in.

Possible Answers:

$121\pi \text{ in}^2$ $\displaystyle 121\pi \text{ in}^2$

$48\pi \text{ in}^2$ $\displaystyle 48\pi \text{ in}^2$

$72\pi \text{ in}^2$ $\displaystyle 72\pi \text{ in}^2$

$144\pi \text{ in}^2$ $\displaystyle 144\pi \text{ in}^2$

$36\pi \text{ in}^2$ $\displaystyle 36\pi \text{ in}^2$

Correct answer:

$144\pi \text{ in}^2$ $\displaystyle 144\pi \text{ in}^2$

Explanation:

To find the surface area of a sphere, we will use the following formula:

$SA = 4\pi r^2$ $\displaystyle SA = 4\pi r^2$

where r is the radius of the sphere.

Now, we know the radius of the sphere is 6in. Knowing this, we can substitute into the formula. We get

$SA = 4 \cdot \pi \cdot (6\text{in})^2$ $\displaystyle SA = 4 \cdot \pi \cdot (6\text{in})^2$

$SA = 4 \cdot \pi \cdot 36\text{in}^2$ $\displaystyle SA = 4 \cdot \pi \cdot 36\text{in}^2$

$SA = 144\pi \text{ in}^2$ $\displaystyle SA = 144\pi \text{ in}^2$

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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 12in.

Possible Answers:

$864\pi \text{ in}^2$ $\displaystyle 864\pi \text{ in}^2$

$144\pi \text{ in}^2$ $\displaystyle 144\pi \text{ in}^2$

$576\pi \text{ in}^2$ $\displaystyle 576\pi \text{ in}^2$

$432\pi \text{ in}^2$ $\displaystyle 432\pi \text{ in}^2$

$216\pi \text{ in}^2$ $\displaystyle 216\pi \text{ in}^2$

Correct answer:

$576\pi \text{ in}^2$ $\displaystyle 576\pi \text{ in}^2$

Explanation:

To find the surface area of a sphere, we will use the following formula:

$SA = 4 \cdot \pi \cdot r^2$ $\displaystyle SA = 4 \cdot \pi \cdot r^2$

where r is the radius of the sphere.

Now, we know the radius of the sphere is 12in.

Knowing this, we can substitute into the formula. We get

$SA = 4 \cdot \pi \cdot (12\text{in})^2$ $\displaystyle SA = 4 \cdot \pi \cdot (12\text{in})^2$

$SA = 4 \cdot \pi \cdot 144\text{in}^2$ $\displaystyle SA = 4 \cdot \pi \cdot 144\text{in}^2$

$SA = 576\pi \text{ in}^2$ $\displaystyle SA = 576\pi \text{ in}^2$

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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 diameter of 20in.

Possible Answers:

$400\pi \text{ in}^2$ $\displaystyle 400\pi \text{ in}^2$

$600\pi \text{ in}^2$ $\displaystyle 600\pi \text{ in}^2$

$200\pi \text{ in}^2$ $\displaystyle 200\pi \text{ in}^2$

$300\pi \text{ in}^2$ $\displaystyle 300\pi \text{ in}^2$

$800\pi \text{ in}^2$ $\displaystyle 800\pi \text{ in}^2$

Correct answer:

$400\pi \text{ in}^2$ $\displaystyle 400\pi \text{ in}^2$

Explanation:

To find the surface area of a sphere, we will use the following formula:

$SA = 4 \cdot \pi \cdot r^2$ $\displaystyle SA = 4 \cdot \pi \cdot r^2$

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 20in. We also know the diameter is two times the radius. Therefore, the radius is 10in.

Knowing this, we can substitute into the formula. We get

$SA = 4 \cdot \pi \cdot (10\text{in})^2$ $\displaystyle SA = 4 \cdot \pi \cdot (10\text{in})^2$

$SA = 4 \cdot \pi \cdot 100\text{in}^2$ $\displaystyle SA = 4 \cdot \pi \cdot 100\text{in}^2$

$SA = 400 \pi \text{ in}^2$ $\displaystyle SA = 400 \pi \text{ in}^2$

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ISEE Upper Level Math : How to find the surface area of a sphere

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #82 : Solid Geometry

Example Question #402 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Example Question #83 : Solid Geometry

Example Question #84 : Solid Geometry

Example Question #5 : How To Find The Surface Area Of A Sphere

Example Question #6 : How To Find The Surface Area Of A Sphere

Example Question #13 : Spheres

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

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 ISEE Upper Level Math Resources

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