Solve by Graphing

Geometry Visualization on the SAT: Desmos Walkthrough

Define each measurement as a variable in Desmos and let it evaluate the geometry formulas step by step, like a slant height as the hypotenuse of a right triangle.

This lesson solves by graphing: turn the equation into curves, then read intersections, vertices, or tangency straight off the Desmos graph.

Take this lesson interactively

A real SAT-style question

A right circular cone has a volume of $71,148 pi$ cubic centimeters and the area of its base is $5,929 pi$ square centimeters. What is the slant height, in centimeters, of this cone?

$12$

$36$

$77$

$85$

How to solve it in Desmos, step by step

The idea

The base area gives the radius and the volume gives the height, then the slant height is the hypotenuse of the right triangle with legs equal to the radius and height. Define each value as a variable in Desmos and let it evaluate.

1
Step 1
Type this to get the radius. The base area pi times r squared equals 5929 pi, so r squared is 5929. Desmos shows r equals 77.
Type in Desmos
$r=\sqrt{5929}$
2
Step 2
Type this to get the height. The volume one third pi r squared h equals 71148 pi gives h equals 3 times 71148 divided by 5929. Desmos shows h equals 36.
Type in Desmos
$h=\frac{3\cdot71148}{5929}$
3
Step 3
Type this for the slant height, the hypotenuse of the triangle with legs r and h. Desmos shows 85, so the answer is D.
Type in Desmos
$\sqrt{r^{2}+h^{2}}$

The answer is D.

$85$

Now you try one

Same question type, different numbers. Use the exact Desmos moves from the walkthrough above.

A right circular cone has a volume of

43{,}008\pi

cubic centimeters and the area of its base is

1{,}024\pi

square centimeters. What is the slant height, in centimeters, of this cone?

Practice this with a live Desmos calculator

The interactive version of this lesson builds the Desmos graph up step by step as you click through, with a live calculator you can type into and your progress saved.

Open the interactive lesson

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