Question type

Arc length and sector area on the SAT

A central angle is a fraction of the full circle, and the arc and the sector are the same fraction of the circumference and the area: arc length $= \frac{θ}{360} \cdot 2 π r$ , sector area $= \frac{θ}{360} \cdot π r^{2}$ . With radius 9 and a 40° angle, the arc is $\frac{40}{360} \cdot 18 π = 2 π$ .

These questions give some of a circle's radius, central angle, arc length and sector area, and ask for another.

It appeared on one of the five official practice tests we analysed. One proportion covers every version.

On 1 of 5 official practice tests
Desmos helps

Why students miss it

Using the area formula for an arc, or the circumference for a sector.

Using the diameter where the radius goes.

Mixing degrees and radians. In radians, the fraction is $\frac{θ}{2 π}$ .

How to spot it

A central angle, often shown in a shaded sector.
"Arc length", "intercepted arc", "area of the sector".
Answers in terms of $π$ .

The method

One proportion

$\frac{angle}{360} = \frac{arc}{2 π r} = \frac{sector}{π r ^{2}}$ .

Arc length asked: fraction × circumference.
Sector area asked: fraction × area.
Angle asked: part ÷ whole × 360.

With Desmos

Desmos handles the arithmetic; set the proportion up first.

For a sector of area 8π in a circle of radius 8: type x/360*64pi=8pi and read x = 45.

Two real questions

From the 1500prep bank, the way the app shows them. Answer first, then open the explanation.

1Question 1 of 2#48Geo

A circle has radius 9. What is the length of an arc of this circle intercepted by a central angle measuring 40°?

Pick an answer, then check it.

Show the answer and explanation

Arc length is

\frac{θ}{360} \cdot 2 π r

. Here

\frac{40}{360} = \frac{1}{9}

and the circumference is

2 π (9) = 18 π

, so the arc length is

\frac{1}{9} (18 π) = 2 π

2Question 2 of 2#42Geo

The shaded sector shown has area $8 π$ , and the radius of the circle is 8. What is the value of x?

Pick an answer, then check it.

Show the answer and explanation

Sector area is

\frac{x}{360} \cdot π r^{2}

. With

r = 8

, the circle's area is

64 π

, so

\frac{x}{360} (64 π) = 8 π

. That gives

\frac{x}{360} = \frac{8 π}{64 π} = \frac{1}{8}

, so

x = \frac{360}{8} = 45

Questions students ask

What is the formula for arc length?

Arc length = (central angle ÷ 360) × 2πr, or angle × r if the angle is in radians.

How do you find the area of a sector?

Sector area = (central angle ÷ 360) × πr². The sector is the same fraction of the circle's area as the angle is of 360°.

Practise this type until it is automatic

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