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Intermediate Geometry : How to find the surface area of a sphere

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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:

$201.1\:m^2$

$202.7\:m^2$

$201.0\:m^2$

$202.3\:m^2$

$201.6\:m^2$

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 #2 : 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:

$4\pi\:units$

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

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

$8\pi\:units$

$2\pi\:units$

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:

$1017.4\:cm^2$

$1017.8\:cm^2$

$1017.9\:cm^2$

$1017.3\:cm^2$

$972.0\:cm^2$

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:

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

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

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

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

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

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 #5 : 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:

$675\pi$

$900\pi$

$13500\pi$

$300\pi$

$225\pi$

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

If the radius of the sphere is one half, what is the surface area?

Possible Answers:

$\frac{1}{4}\pi$

$\frac{1}{2}\pi$

$\pi$

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

$2\pi$

Correct answer:

$\pi$

Explanation:

Write the formula for the surface area of a sphere, plug in the radius, and find the area.

$A=4\pi r^2 = 4\pi \left(\frac{1}{2}\right)^2 = 4\pi\left(\frac{1}{4}\right)= \pi$

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Example Question #7 : How To Find The Surface Area Of A Sphere

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

Possible Answers:

$900\pi$

$56.25\pi$

$500\pi$

$42\pi$

$225\pi$

Correct answer:

$225\pi$

Explanation:

The radius is half the length of a diameter.

$r=\frac{d}{2}=\frac{15}{2}=7.5$

Write the formula for the surface area of a sphere and plug in the radius.

$A=4\pi r^2 = 4\pi(7.5)^2= 225\pi$

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

Find the surface area of a sphere if the volume is .

Possible Answers:

$\left(\frac{3}{4\pi}\right)^\frac{2}{3}$

$\sqrt[3]{9}$

$4\pi\left(\frac{4}{3\pi}\right)^\frac{2}{3}$

$\sqrt[3]{\frac{3\pi}{4}}$

$4\pi\left(\frac{3}{4\pi}\right)^\frac{2}{3}$

Correct answer:

$4\pi\left(\frac{3}{4\pi}\right)^\frac{2}{3}$

Explanation:

Write the formula for the volume of a sphere and substitute volume to find radius.

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

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

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

$r=\sqrt[3]{\frac{3}{4\pi}}= (\frac{3}{4\pi})^\frac{1}{3}$

Write the formula for the surface area of a sphere and plug in the radius.

$A=4\pi r^2 = 4\pi \left[\left(\frac{3}{4\pi}\right)^\frac{1}{3}\right]^2 = 4\pi\left(\frac{3}{4\pi}\right)^\frac{2}{3}$

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Example Question #9 : How To Find The Surface Area Of A Sphere

Find the surface area of a hemisphere that has a radius of .

Possible Answers:

37.70

32.60

39.60

38.80

Correct answer:

37.70

Explanation:

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}=4\pi r^2$ , where is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}=\pi r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}=2\pi r^2+\pi r^2=3 \pi r^2$

Plug in the given radius to find the surface area.

$\text{Surface Area of Hemisphere}=3\pi (2)^2=37.70$

Make sure to round to places after the decimal.

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

Find the surface area of a hemisphere that has a radius of .

Possible Answers:

84.82

79.40

86.08

85.22

Correct answer:

84.82

Explanation:

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}=4\pi r^2$ , where is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}=\pi r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}=2\pi r^2+\pi r^2=3 \pi r^2$

Plug in the given radius to find the surface area.

$\text{Surface Area of Hemisphere}=3\pi (3)^2=84.82$

Make sure to round to places after the decimal.

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Intermediate Geometry : How to find the surface area of a sphere

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

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

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

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

Example Question #7 : How To Find The Surface Area Of A Sphere

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

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

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