ISEE Upper Level Quantitative Reasoning - How to find the surface area of a sphere

Find the surface area of a sphere with a diameter of 20in.

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

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

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

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

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

Explanation

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

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

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

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

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

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

Give the surface area of a sphere with diameter .

$64 \pi$

$32 \pi$

$128 \pi$

$256 \pi$

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

Explanation

A sphere with diameter has radius half that, or , so substitute into the formula for the surface area of a sphere:

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

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

$100\pi m^2$

$50\pi m^2$

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:

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

Plug in our radius and solve!

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

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?

$576\pi in^2$

$144 \pi in^2$

$192\pi in^2$

Not enough information provided

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:

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

Now, just plug in our known radius and simplify:

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

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

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

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

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

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

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

Explanation

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

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

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

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

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

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

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

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

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

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

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

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

Explanation

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

$SA = 4\pi r^2$

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

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

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

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

Find the surface area of a sphere with a radius of 12in.

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

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

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

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

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

Explanation

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

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

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

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

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

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

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

$100\pi m^2$

$50\pi m^2$

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:

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

Plug in our radius and solve!

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

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

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

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

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

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

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

Explanation

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

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

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

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

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

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

Find the surface area of a sphere with a diameter of 12in.

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

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

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

$108\pi \text{ in}^2$

$96\pi \text{ in}^2$

Explanation

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

$SA = 4\pi r^2$

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 12in. We also know the diameter is two times the radius. Therefore, the radius is 6in. So, we get

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

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

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

How to find the surface area of a sphere

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