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

In terms of $\pi$ , give the surface area, in square inches, of a spherical water tank with a diameter of 20 feet.

$57,600 \pi\textrm{ in}^{2}$

$2,304,000 \pi \textrm{ in}^{2}$

$18,432,000\pi\textrm{ in}^{2}$

$230,400 \pi\textrm{ in}^{2}$

$307,200\pi\textrm{ in}^{2}$

Explanation

feet = $20 \times 12 = 240$ inches, the diameter of the tank; half of this, or 120 inches, is the radius. Set , substitute in the surface area formula, and solve for :

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

$A =4\pi \cdot 120^{2}$

$A =4\pi \cdot 14,400$

$A = 57,600 \pi$

is a positive number. Which is the greater quantity?

(A) The surface area of a sphere with radius

(B) The surface area of a cube with edges of length

(B) is greater

(A) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

Explanation

The surface area of a sphere is $4 \pi$ times the square of its radius, which here is ; the surface area of the sphere in (A) is $4 \pi t^{2}$ .

The area of one face of a cube is the square of the length of an edge, which here is , so the area of one face of the cube in (B) is $(2t) ^{2} = 4t^{2}$ . The cube has six faces so the total surface area is $6 \cdot 4t^{2} = 24t^{2}$ .

$\pi < 4$ , so $4 \pi t^{2} < 16 t^{2} < 24 t^{2}$ , giving the sphere less surface area. (B) is greater.

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:

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

Sphere A has volume $972 \pi \textrm{ in}^{3}$ . Sphere B has surface area $400 \pi \textrm{ in}^{2}$ . Which is the greater quantity?

(a) The radius of Sphere A

(b) The radius of Sphere B

(b) is greater

(a) is greater

(a) and (b) are equal

It is impossible to tell from the information given

Explanation

(a) Substitute $V = 972 \pi$ in the formula for the volume of a sphere:

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

$\frac{4}{3}\pi r^{3}= 972 \pi$

$\frac{4}{3}\pi r^{3} \div \frac{4}{3}\pi = 972 \pi \div \frac{4}{3}\pi$

$\frac{4}{3}\pi r^{3} \cdot \frac{3}{4\pi} = 972 \pi\cdot \frac{3}{4\pi}$

$r^{3} = 729$

$r =\sqrt[3]{ 729} = 9$ inches

(b) Substitute $A = 400 \pi$ in the formula for the surface area of a sphere:

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

$4\pi r^{2}= 400 \pi$

$4\pi r^{2} \div 4\pi = 400 \pi \div 4\pi$

$r^{2} = 100$

$r = \sqrt{100} = 10$ inches

(b) is greater.

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 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 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 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

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

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

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

How to find the surface area of a sphere

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