SAT Volume and Surface Area on Desmos: Cylinders, Cones

To solve an SAT volume problem on Desmos, pick the formula, give each measurement its own line, and type the formula on the last line. Desmos does the arithmetic and shows the result. You can type every line on this page into the College Board version of Desmos on our Desmos on the SAT practice page.

Bluebook has a Desmos calculator built in, and you can use it at any point in the Math section. Bluebook also shows a reference sheet of common formulas on tests with math questions. This page gives you the formulas as plain math, a section on cylinder surface area, one worked pipe problem, one guided cone problem, and three practice problems with answers.

Volume formulas for SAT solids

Each formula uses r for the radius and h for the height. Check whether the question gives a radius or a diameter before you type anything.

A few keystrokes cover every line on this page. Type pi and Desmos writes π, and type * for the multiplication dot. Type / to start a fraction, ^ for an exponent and _ for a subscript, then press the right arrow to leave it.

Cylinder surface area: lateral and total

The lateral surface area of a cylinder is 2πrh. That is the curved side only, like the label on a can. Unroll the label and you get a rectangle. One side is the circle's circumference, 2πr, and the other side is the height, h.

The total surface area adds the two circular ends, each πr², so it is 2πr² + 2πrh. Factor out 2πr and you get the short form, 2πr(h + r).

On Desmos, type 2π·3·8 for the lateral area and 2π·3(8 + 3) for the total. The lines show 150.796... and 207.345... Then read the wording again. "Lateral" or "curved surface" means the side only. "Total surface area" means the side plus both ends. For a sphere, the surface area is 4πr².

Worked example: a hollow pipe (I do)

A hollow steel pipe is a right circular cylinder with an outside radius of 24 inches, a wall thickness of 5/8 inch, and a height of 144 inches. Which choice is closest to the volume of the steel wall, in cubic inches?

A) 177
B) 565
C) 2,160
D) 13,395

SAT-style question written for this page.

Pick the formula. The steel fills the space between two cylinders. So the wall volume is the outer cylinder minus the inner one: πh(R² - r²).

Find the inner radius. The wall is 5/8 inch thick, so the inner radius is 24 - 5/8 = 23.375 inches.

1  R = 24
2  r_1 = R - 5/8
3  h = 144
4  πh(R^2 - r_1^2)

Line 2 shows 23.375, and line 4 shows 13394.9656767. The closest choice is D) 13,395.

We gave the inner radius a subscript on purpose. When we typed plain r = R - 5/8, Desmos also drew a graph for that line. With r_1, the line stays a plain number.

Check by hand. R² - r² factors as (R - r)(R + r) = 0.625 × 47.375 = 29.609375. Multiply by 144 to get 4,263.75, and 4,263.75π ≈ 13,394.97. That matches D.

Guided example: a cone with the top cut off (we do)

A cone has a base radius of 6 cm and a height of 10 cm. A cut parallel to the base removes the top 5 cm of the cone, which is a smaller cone. What is the volume of the piece that remains?

SAT-style question written for this page.

Step 1 (shown). The small cone on top is similar to the whole cone. Its height is 5 cm, half of 10, so its radius is half of 6: 3 cm.

Step 2 (shown). Put each cone on its own line.

1  V_1 = (1/3)π·6^2·10
2  V_2 = (1/3)π·3^2·5

Line 1 shows 376.991118431, and line 2 shows 47.1238898038.

Step 3 (you finish). Subtract the small cone from the whole cone on line 3. If the choices are written with π, add a line 4 that divides your line 3 by π.

Check your answer: V_1 - V_2 shows 329.867228627, so the remaining piece is about 329.87 cm³. (V_1 - V_2)/π shows 105, so the exact answer is 105π cm³. By hand: 120π - 15π = 105π.

The small cone holds 15π out of 120π, which is 1/8 of the whole cone. Halving every length multiplies the volume by (1/2)³ = 1/8.

Your turn: three SAT-style practice problems (you do)

We wrote these three for this page. Solve each one on Desmos first. The answers are at the end of this section.

Practice 1. A sphere has a surface area of 100π square centimeters. What is its volume, in cubic centimeters?

Practice 2. A water tank is a cylinder with radius 4 meters and height 9 meters, with a hemisphere of radius 4 meters on top. To the nearest cubic meter, what is the tank's total volume?

Practice 3. A closed cylinder has radius 5 inches and height 12 inches. What are its lateral surface area and its total surface area, in terms of π?

Answers

1. 500π/3, about 523.6 cm³. Set 4πr² = 100π and divide both sides by 4π: r² = 25, so r = 5. Then (4/3)π·5³ = 500π/3. To let Desmos find r, type 4πx^2 = 100π. It graphs two vertical lines, at x = -5 and x = 5, and a radius is the positive one. If you type the same equation with r_1 in place of x, Desmos shows an error instead, because it only graphs equations in x and y.
2. 586 m³. The cylinder is π·4²·9 = 144π ≈ 452.39. The hemisphere is (2/3)π·4³ = 128π/3 ≈ 134.04. The total is 560π/3 ≈ 586.43, which rounds to 586. One Desmos line does it: π·4^2·9 + (2/3)π·4^3.
3. Lateral 120π ≈ 376.99 in²; total 170π ≈ 534.07 in². Lateral: 2π·5·12 = 120π. Total: 2π·5(12 + 5) = 170π. The two ends add 2π·5² = 50π.

What happens to volume when the radius doubles

A cylinder's or cone's volume depends on r². If the radius doubles and the height stays the same, the volume is multiplied by 2² = 4. If the radius and the height both double, the volume is multiplied by 2³ = 8. A sphere's volume depends on r³, so doubling its radius multiplies its volume by 8.

To check a scaling claim on Desmos, define the formula as a function and compare two versions:

1  V(r, h) = πr^2h
2  V(6, 5)/V(3, 5)
3  V(6, 10)/V(3, 5)

Line 2 shows 4, and line 3 shows 8.

When algebra is faster

College Board's calculator policy says some Math questions are better done without a calculator, even though you're allowed to use one. Scaling questions are the clearest case: 2² = 4 takes less time than three lines of Desmos. Finding the sphere's radius in Practice 1 is another. Dividing by 4π is one step.

Mistakes to avoid

• Typing the diameter as the radius. If a cylinder has a diameter of 10, type 5. Using 10 in πr²h makes the volume 4 times too big.
• Using only the outer radius of a hollow object. In the pipe problem, π(24²)(144) ≈ 260,576 cubic inches. That is the volume of a solid rod, not the steel wall.
• Mixing up lateral and total surface area. 2πrh leaves out both ends, and 2πr(h + r) includes them. For r = 3 and h = 8, that is 48π against 66π.

Practice next

Work through the free Area, Perimeter and Volume lesson notes. Then try more geometry sets in the TTA SAT question bank.

When a volume question is wrapped in a story, set it up the way you would any other word problem: see SAT word problems on Desmos. For more geometry on Desmos, try circle tangent slope with a Desmos check and parallel lines and angles on Desmos.

Checked against College Board's calculator policy and Bluebook testing tools page, with every Desmos line typed into Desmos's College Board graphing calculator, on September 26, 2026. The Test Advantage is independent of College Board and Desmos.