Geometry Arc Length and Sector Area Lesson (Free PDF)

Geometry math lesson: Arc Length and Sector Area – Maths PDF

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This arcs sectors lesson for Geometry covers arc length and sector area (Common Core G-C.B.5), with a free PDF to print.

Arc length is the fraction theta/360 of the circumference, so s = (theta/360)(2 pi r). Sector area is the same fraction of the circle’s area, A = (theta/360)(pi r squared). One radian is the angle whose arc equals the radius, so 180 degrees equals pi radians. In radians, s = r theta.

Geometry · Common Core State Standards: G-C.B.5

Key facts in this arcs sectors lesson

Key facts

Arc length = (angle/360) x 2 pi r.
Sector area = (angle/360) x pi r squared.
180 degrees = pi radians.
In radians: arc length s = r times theta, area A = (1/2) r squared theta.
A radian is the central angle whose arc length equals the radius.

How do you find arc length?

An arc is a fraction of the whole circle. A central angle of \(\text{theta}\) degrees is \(\frac{\text{theta}}{360}\) of a full turn, so its arc length is that fraction of the circumference.

\[s = \frac{\text{theta}}{360} \cdot 2\pi r\]

Arc length = (60/360) of the circumference = (1/6)(2π · 6) = 2π.

For \(r = 6\) and \(\text{theta} = 60^{\circ}\): \(s = \frac{60}{360} \cdot 2\pi \cdot 6 = 2\pi = 6.28\) units, which is about 6.28.

How do you find the area of a sector?

A sector is the region between two radii and an arc. Its area is the same fraction of the circle’s area.

\[A = \frac{\text{theta}}{360} \cdot \pi r^{2}\]

Sector area = (90/360) of the circle’s area = 25π.

A pizza cut into 8 equal slices: each slice is a sector of 45°.

What is a radian?

One radian is the measure of a central angle whose arc has the same length as the radius. A full circle is \(2\pi\) radians, so \(180^{\circ} = \pi\) radians. In radian measure, arc length and sector area are simple: \(s = r\text{theta}\) and \(A = \frac{1}{2}r^{2}\text{theta}\).

One radian: the arc has the same length as the radius.

Degrees	Radians
\(30^{\circ}\)	\(\frac{\pi}{6}\)
\(45^{\circ}\)	\(\frac{\pi}{4}\)
\(60^{\circ}\)	\(\frac{\pi}{3}\)
\(90^{\circ}\)	\(\frac{\pi}{2}\)
\(180^{\circ}\)	\(\pi\)

To convert, multiply degrees by \(\frac{\pi}{180}\) to get radians, or multiply radians by \(\frac{180}{\pi}\) to get degrees.

How do you solve an arc or sector problem?

Write the fraction of the circle: angle divided by 360 (or by 2pi).
Multiply by the circumference for arc length or by the area for a sector.
Keep pi in the answer or round at the end, and include units (square units for area).

Example: A sector with radius 8 and central angle \(45^{\circ}\) has area \(\frac{45}{360} \cdot \pi \cdot 8^{2} = 8\pi\), about 25.13 square units.

Arcs sectors lesson: questions and answers

How do you find the length of an arc?

Multiply the circumference by the fraction of the circle: angle divided by 360. For radius 6 and 60 degrees the arc length is 2 pi, about 6.28.

What is a sector?

A sector is the region bounded by two radii and the arc between them, like a slice of pizza. Its area is the same fraction of the circle’s area as its angle is of 360 degrees.

What is a radian?

A radian is the angle at the center of a circle that intercepts an arc equal in length to the radius. About 57.3 degrees make one radian.

How do you convert degrees to radians?

Multiply by pi over 180. For example, 135 degrees is 3 pi over 4 radians.

Why is the area of a sector one half r squared theta?

In radians the fraction of the circle is theta over 2 pi, so the area is (theta/2pi)(pi r squared) = (1/2) r squared theta.

How to use this arcs sectors lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

Download the arcs sectors lesson as a PDF with the button below and keep it with your notes.

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