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Digital SAT · Solve it in Desmos

How to Solve Arc Length and Sector Area on Desmos (Digital SAT)

Desmos carries the pi and the numbers, so it shines on grid-ins where rounding early would cost the answer. Getting the fraction and the units right is the part it leaves to you.

Perfect1600 Content Team·Uses the same Desmos calculator that is built into the Digital SAT

Full method for arc length and sector area

The Desmos steps

  1. 1

    Type the fraction times the whole

    For an arc, something like (pi/4)/(2 pi) * 2 pi * 8, all on one line.

  2. 2

    Keep pi exact

    Enter pi directly so nothing gets rounded early.

  3. 3

    Reverse for a missing piece

    For a given arc or sector, type the equation and read the angle or radius.

Open a real question in the same Desmos calculator you get on Perfect1600, with the setup already typed in:

Worked in Desmos

Medium example
In a circle of radius 8, a central angle of π4\frac{\pi}{4} radians subtends an arc. What is the length of that arc?
2π2\pi
B
4π4\pi
C
8π8\pi
D
16π16\pi

A: Correct. Arc length =rθ=8×π4=2π= r\theta = 8 \times \frac{\pi}{4} = 2\pi.

B: Incorrect. This uses θ=π2\theta = \frac{\pi}{2} instead of π4\frac{\pi}{4}.

C: Incorrect. This multiplies by π\pi rather than π4\frac{\pi}{4}.

D: Incorrect. This uses the diameter and π\pi.

Explanation

Arc length =rθ=8×π4=2π= r\theta = 8 \times \frac{\pi}{4} = 2\pi.

Medium example
In a circle of radius 10, an arc has length 5π5\pi. What is the measure of the central angle that subtends this arc, in degrees?
A
45
B
60
90
D
120

A: Incorrect. This corresponds to an arc of 2.5π2.5\pi.

B: Incorrect. This corresponds to a fraction of 16\frac{1}{6}.

C: Correct. The arc is 5π20π=14\frac{5\pi}{20\pi} = \frac{1}{4} of the circle, so the angle is 90∘90^\circ.

D: Incorrect. This corresponds to a fraction of 13\frac{1}{3}.

Explanation

The circumference is 20π20\pi; 5π20π=14\frac{5\pi}{20\pi} = \frac{1}{4}, so the angle is 90∘90^\circ.

Medium example
In a circle of radius 6, a sector has area 12π12\pi. What is the measure of the central angle of this sector, in degrees?
A
45
B
60
C
90
120

A: Incorrect. This corresponds to a fraction of 18\frac{1}{8}.

B: Incorrect. This corresponds to a fraction of 16\frac{1}{6}.

C: Incorrect. This corresponds to a fraction of 14\frac{1}{4}.

D: Correct. The sector is 12π36π=13\frac{12\pi}{36\pi} = \frac{1}{3} of the circle, so the angle is 120∘120^\circ.

Explanation

The area is 36π36\pi; 12π36π=13\frac{12\pi}{36\pi} = \frac{1}{3}, so the angle is 120∘120^\circ.

Try it with the calculator

Question 1easy
Circle sector
120°

A circle has area 36. A central angle of 120∘120^\circ defines a sector. What is the area of that sector?

Question 2easy
Circle sector
60°

A circle has circumference 30. A central angle of 60∘60^\circ subtends an arc. What is the length of that arc?

Question 3 is ready when you are

Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.

Start the free diagnostic

Desmos questions

Can Desmos solve arc length and sector area questions on the SAT?

Desmos carries the pi and the numbers, so it shines on grid-ins where rounding early would cost the answer. Getting the fraction and the units right is the part it leaves to you.

Is the Desmos calculator really built into the Digital SAT?

Yes. The Bluebook testing app includes the Desmos graphing calculator on every Math question, so the method here is one you can use on test day, not a workaround.

Should I still learn the algebra?

Yes. Desmos is fastest when you know what to type and what to read. Learn the underlying method on the full guide, then use the calculator to move quickly and to check your work.

Master arc length and sector area, calculator and all

Every Digital SAT question type comes with the method, the Desmos shortcut, and free practice with instant feedback.

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