Circumference & Arc Length

Q: What if the angle is given in radians?

Radians make arc length even simpler: $s = r\theta$, with no fraction at all. That works because a radian is defined so a full turn is $2\pi$ radians. The sector area version is $A = \dfrac{1}{2}r^2\theta$. If the angle is in degrees, stick with the $\dfrac{\theta}{360}$ formulas.

An arc is a piece of a circle's edge, and a sector is a pizza-slice piece of its inside. Both are measured with the same trick: figure out what fraction of the whole circle you have, then take that fraction of the whole circle's measurement. The fraction always comes from the central angle — the angle at the center between the two radii — divided by $360^\circ$ .

That's the entire topic. Arc length is a fraction of the circumference $2\pi r$ , and sector area is the same fraction of the area $\pi r^2$ . If you can find the fraction and you know the two circle formulas, every arc and sector problem is two lines of work.

One idea: the fraction of the circle

A central angle of $\theta$ degrees cuts off $\dfrac{\theta}{360}$ of the circle. In the figure below, the $120^\circ$ angle claims $\dfrac{120}{360} = \dfrac{1}{3}$ of the circle — one third of the edge and one third of the inside. Reduce the fraction before doing anything else; it keeps the arithmetic small.

The two radii and the angle between them define both the arc (the curved edge between $A$ and $B$ ) and the sector (the slice enclosed by the two radii and that arc).

Arc length: a fraction of the circumference

Arc length is a distance along the edge, so it comes from the circumference: $\ell = \dfrac{\theta}{360} \cdot 2\pi r$ . A $90^\circ$ arc is a quarter of the circumference, a $60^\circ$ arc is a sixth, and so on.

Don't confuse arc length with arc measure. Arc measure is just the central angle in degrees — a $60^\circ$ arc has measure $60^\circ$ in a circle of any size. Arc length is an actual distance, and it grows with the radius: a $60^\circ$ arc on a bigger circle is longer.

Sector area: the same fraction of the area

Sector area comes from the circle's area: $A = \dfrac{\theta}{360} \cdot \pi r^2$ . Same fraction, different base formula.

That's also how to keep the two formulas straight: length questions use the length formula $2\pi r$ , and area questions use the area formula $\pi r^2$ . If your sector answer doesn't involve $r^2$ somewhere, you likely used the circumference by mistake.

Worked examples

Example 1: arc length

A circle has a radius of $6$ . Find the length of an arc with a central angle of $60^\circ$ .

Write the arc length formula

\ell = \dfrac{\theta}{360} \cdot 2\pi r

Substitute and reduce the fraction

\ell = \dfrac{60}{360} \cdot 2\pi(6) = \dfrac{1}{6} \cdot 12\pi

Simplify

\ell = 2\pi

Answer: $\ell = 2\pi \approx 6.3$

Example 2: arc length from a diameter

A circle has a diameter of $10$ . Find the length of an arc with a central angle of $144^\circ$ .

Halve the diameter to get the radius

r = \dfrac{10}{2} = 5

Find the fraction of the circle

\dfrac{144}{360} = \dfrac{2}{5}

Multiply by the circumference

\ell = \dfrac{2}{5} \cdot 2\pi(5) = \dfrac{2}{5} \cdot 10\pi

Simplify

\ell = 4\pi

Answer: $\ell = 4\pi \approx 12.6$

Example 3: sector area

A circle has a radius of $6$ . Find the area of a sector with a central angle of $120^\circ$ .

Write the sector area formula

A = \dfrac{\theta}{360} \cdot \pi r^2

Substitute and reduce the fraction

A = \dfrac{120}{360} \cdot \pi(6)^2 = \dfrac{1}{3} \cdot 36\pi

Simplify

A = 12\pi

Answer: $A = 12\pi \approx 37.7$ square units

Example 4: working backward to the angle

A sector of a circle with radius $6$ has an area of $6\pi$ . Find its central angle.

Set up the sector area formula

6\pi = \dfrac{\theta}{360} \cdot \pi(6)^2

Simplify the right side

6\pi = \dfrac{\theta}{360} \cdot 36\pi

Divide both sides by

36\pi

\dfrac{\theta}{360} = \dfrac{6\pi}{36\pi} = \dfrac{1}{6}

Multiply both sides by

360

\theta = 60^\circ

Answer: $\theta = 60^\circ$

Try one yourself

Common questions

What's the difference between arc measure and arc length?

Arc measure is the central angle in degrees — it ignores the size of the circle. Arc length is a real distance along the edge, so it depends on the radius: $\ell = \dfrac{\theta}{360} \cdot 2\pi r$ . Two arcs can have the same measure but very different lengths.

How do I remember which formula uses

2\pi r

and which uses

\pi r^2

Match the type of answer. Arc length is a distance, so it takes a fraction of the distance around the circle, $2\pi r$ . Sector area is an area, so it takes the same fraction of $\pi r^2$ . Both fractions are $\dfrac{\theta}{360}$ .

What if the angle is given in radians?

Radians make arc length even simpler: $s = r\theta$ , with no fraction at all. That works because a radian is defined so a full turn is $2\pi$ radians. The sector area version is $A = \dfrac{1}{2}r^2\theta$ . If the angle is in degrees, stick with the $\dfrac{\theta}{360}$ formulas.

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Circumference & Arc Length

One idea: the fraction of the circle

Arc length: a fraction of the circumference

Sector area: the same fraction of the area

Worked examples

Example 1: arc length

Example 2: arc length from a diameter

Example 3: sector area

Example 4: working backward to the angle

Try one yourself

Common questions

Keep learning

Area of Circles

Sine, Cosine & Tangent Ratios

Volume of Prisms & Cylinders