SAT Math · Geometry and Trigonometry

Surface area of a sphere

Q: Is the surface area of a sphere on the SAT reference sheet?

No. The SAT reference sheet in Bluebook gives the volume of a sphere, V = 4/3πr³, but not the surface area. You need to remember SA = 4πr² yourself.

Q: What is the surface area of a hemisphere?

The curved surface of a hemisphere is 2πr², half of a full sphere. If the hemisphere is solid, add the flat circular base, πr², for a total of 3πr².

Q: How do you find the radius of a sphere from its surface area?

Divide the surface area by 4π and take the square root: r = √(SA ÷ 4π). For a surface area of 100π, r² = 25, so the radius is 5.

Q: Why is the surface area of a sphere 4πr²?

Archimedes showed that a sphere's surface has the same area as the curved side of the cylinder that fits around it. That side has area 2πr × 2r, which is 4πr².

Q: How can Aristotle help with SAT geometry?

Aristotle is a voice AI SAT tutor. Your child talks through a geometry question out loud, and Aristotle asks guiding questions instead of giving the answer, starting from the step where their reasoning broke. It tracks 113 SAT Math skills, including area and volume.

The one sphere formula the SAT reference sheet leaves out, and how to use it.

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Updated October 30, 2026

$SA = 4\pi r^2$

The dashed equator is a circle with area $\pi r^2$ . The surface is four of those.

What is the surface area of a sphere?

SA = 4\pi r^2

The surface area is the size of the sphere's skin in square units, like an orange peel laid flat. It depends only on the radius, and because it is an area, the radius is squared.

The skin of a sphere of radius r.

Why four: wrap it in a can

\underbrace{2\pi r}_{\text{around}} \times \underbrace{2r}_{\text{tall}} = 4\pi r^2

Fit a cylinder snugly around the sphere: radius $r$ , height $2r$ . Archimedes proved the sphere's skin has exactly the area of the can's curved side. Unroll that side into a rectangle, $2\pi r$ around by $2r$ tall, and the four appears.

The sphere's skin equals the label on the can: 2πr by 2r.

Half a sphere has a new face

\underbrace{2\pi r^2}_{\text{dome}} + \underbrace{\pi r^2}_{\text{flat disk}} = 3\pi r^2

Cut the sphere in half and the dome keeps half the skin, $2\pi r^2$ . The cut exposes a new flat face, a circle of area $\pi r^2$ . A solid hemisphere has both; a bowl with no lid has only the dome.

Dome 2πr² plus the flat disk πr².

Why it is four circles

Answer each question in your head, then scroll for the answer.

Think it through

Question 1
Wrap a cylinder snugly around a sphere. What are its radius and height?
Answer
Radius $r$ and height $2r$ , the sphere's diameter.
Question 2
What is the area of that cylinder's curved side?
Answer
$2\pi r \times 2r = 4\pi r^2$ . Archimedes showed the sphere's surface is exactly that.
Question 3
Double the radius. What happens to the surface area?
Answer
It becomes 4 times as big, because $r$ is squared.

Try it yourself

Work it out first, then check your answer. SAT-style questions, not official College Board questions.

Question 1

A sphere has a radius of 3 inches. What is its surface area, in square inches?

Question 2

The surface area of a sphere is $100\pi$ square centimeters. What is the radius of the sphere, in centimeters?

Question 3

A solid hemisphere has a radius of 6. What is its total surface area, including the flat base?

Question 4

The radius of a sphere is multiplied by 3. By what factor does its surface area increase?

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The formulas

Sphere surface area

$SA = 4\pi r^2$

$r$ is the radius. The answer is in square units.

Hemisphere, total surface area

$SA = 3\pi r^2$

Half the sphere's curved surface ( $2\pi r^2$ ) plus the flat circular base ( $\pi r^2$ ).

Radius from surface area

$r = \sqrt{\dfrac{SA}{4\pi}}$

Divide the surface area by $4\pi$ , then take the square root.

Sphere volume, for comparison

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

On the reference sheet. Volume uses $r^3$ ; surface area uses $r^2$ .

Try it

Grow the sphere

Change the radius and compare surface area with volume.

Radius r3

Surface areaSA = 4π × 3² = 36π ≈ 113.1

VolumeV = 4/3π × 3³ = 36π ≈ 113.1

Surface area grows with r², volume with r³. At r = 3 they are equal: 36π each.

Common mistakes

Mixing it up with the volume formula

$4\pi r^2$ is surface area, $\tfrac{4}{3}\pi r^3$ is volume. Check the units asked for.

Using the diameter as the radius

A diameter of 10 means $r = 5$ . Using 10 makes it four times too big.

Forgetting the base of a hemisphere

A solid hemisphere has a flat base too: $3\pi r^2$ in total, not $2\pi r^2$ .

Stopping at r²

Working backward gives $r^2$ first. Take the square root.

How it shows up on the SAT

Geometry and Trigonometry is about 15% of SAT Math. Expect to find the radius from the surface area, handle a hemisphere, or scale the radius.

What Aristotle tracks in area and volume

5 SAT Math skills, including:

Finding area and perimeter from given information
Finding surface area and volume from given information
Using scale factors for length, area, and volume
Choosing and using area formulas accurately

What is the formula for the surface area of a sphere?

SA = 4πr², where r is the radius of the sphere. It equals four times the area of a circle with the same radius, and the answer is in square units.

Is the surface area of a sphere on the SAT reference sheet?

What is the surface area of a hemisphere?

How do you find the radius of a sphere from its surface area?

Why is the surface area of a sphere 4πr²?

How can Aristotle help with SAT geometry?

More SAT concepts

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