Sector of a Circle: Area, Arc Length, Perimeter

What Is a Sector of a Circle?

A sector of a circle is the region bounded by two radii and the arc between them, the pie-slice or pizza-slice shape. The angle between the two radii, measured at the centre, is the sector's central angle, written θ.

Every sector comes in a pair. The smaller slice, with a central angle less than $180°$, is the minor sector; the larger piece, with an angle greater than $180°$, is the major sector. Together they make the whole circle. A sector is easy to confuse with a segment: a sector is bounded by two straight radii and an arc, while a segment is bounded by a straight chord and an arc. Two radii means sector; one chord means segment.

The Sector Formulas — Area, Arc Length, and Perimeter

Every sector formula is the same single idea: a sector is the fraction $\dfrac{\theta}{360°}$ of the whole circle (in degrees), so it takes that same fraction of the circle's area and of its circumference. Define the variables once: r is the radius, θ the central angle, l the arc length (the curved edge of the sector), and π ≈ 3.14159.

Area of a sector:

The whole circle's area is $\pi r^2$, and the sector is the angle's fraction of it:

$$\text{Area} = \frac{\theta}{360°} \times \pi r^2 \quad (\theta \text{ in degrees}).$$

In radians, a full circle is $2\pi$, so the fraction becomes $\dfrac{\theta}{2\pi}$, and the area simplifies neatly:

$$\text{Area} = \frac{1}{2} r^2 \theta \quad (\theta \text{ in radians}).$$

Arc length:

The arc is the same fraction of the full circumference $2\pi r$:

$$l = \frac{\theta}{360°} \times 2\pi r \quad (\theta \text{ in degrees}), \qquad l = r\theta \quad (\theta \text{ in radians}).$$

Area without the angle:

If you know the arc length $l$ and radius $r$ but not the angle, the area is:

$$\text{Area} = \frac{1}{2} l , r.$$

This comes from substituting $l = r\theta$ into the radian area formula, so it is not a new rule, just the same one rearranged. (The full treatment of the curved edge on its own is in arc length.)

Perimeter of a sector. The boundary of a sector is two straight radii plus the curved arc, so:

$$\text{Perimeter} = 2r + l.$$

Examples of the Sector of a Circle

With the fraction idea and the formulas in hand, here is the sector doing real work, from a one-step area up to a problem that finds the angle from a known area. The problems build from clean degrees to radians and a real-world slice.

Example 1 - A sector of a circle has a radius of 6 cm and a central angle of $60°$. Find its area. Use $\pi = 3.14$.

The sector is $\dfrac{60}{360} = \dfrac{1}{6}$ of the circle:

$$\text{Area} = \frac{\theta}{360°} \times \pi r^2 = \frac{60}{360} \times 3.14 \times 6^2 = \frac{1}{6} \times 113.04 = 18.84 \text{ cm}^2.$$

Final answer: 18.84 cm².

Example 2 - A sector has a radius of 7 cm and an arc length of 11 cm. Find its area. Use $\pi = \dfrac{22}{7}$.

A first instinct is to reach for $\dfrac{\theta}{360°}\times\pi r^2$, but no angle is given, so a student often tries to multiply the arc length by the radius and call that the area: $11 \times 7 = 77$ cm². Check that against the picture. That product, arc times radius, covers a region the size of a rectangle 11 cm by 7 cm, but the sector is a thin curved wedge that fills only half of such a rectangle. The figure $77$ is too big by a factor of two.

The correct formula for the area from the arc length is $\dfrac{1}{2} l,r$, which is exactly half of that product:

$$\text{Area} = \frac{1}{2} l , r = \frac{1}{2} \times 11 \times 7 = 38.5 \text{ cm}^2.$$

Final answer: 38.5 cm².

Example 3 - A sector has a radius of 5 m and a central angle of 2 radians. Find its area and arc length.

In radians, the area is $\tfrac{1}{2}r^2\theta$ and the arc length is $r\theta$:

$$\text{Area} = \frac{1}{2} \times 5^2 \times 2 = 25 \text{ m}^2, \qquad l = r\theta = 5 \times 2 = 10 \text{ m}.$$

Final answer: area 25 m², arc length 10 m.

Example 4 - Find the perimeter of a sector of radius 10 cm whose central angle is $72°$. Use $\pi = 3.14$.

First find the arc length, then add the two radii:

$$l = \frac{72}{360} \times 2 \times 3.14 \times 10 = \frac{1}{5} \times 62.8 = 12.56 \text{ cm}.$$

$$\text{Perimeter} = 2r + l = 2 \times 10 + 12.56 = 32.56 \text{ cm}.$$

Final answer: 32.56 cm.

Example 5 - A sector of a circle of radius 12 cm has an area of $48\pi$ cm². Find its central angle in degrees.

Set the area formula equal to the given area and solve for θ:

$$\frac{\theta}{360°} \times \pi \times 12^2 = 48\pi ;\Rightarrow; \frac{\theta}{360°} \times 144 = 48 ;\Rightarrow; \theta = \frac{48}{144} \times 360° = 120°.$$

Final answer: $\theta = 120°$.

Example 6 - A windscreen wiper of length 25 cm sweeps through an angle of $108°$. Find the area of the windscreen it cleans. Use $\pi = 3.14$.

The wiper traces a sector with radius 25 cm and central angle $108°$:

$$\text{Area} = \frac{108}{360} \times 3.14 \times 25^2 = 0.3 \times 3.14 \times 625 = 588.75 \text{ cm}^2.$$

Final answer: 588.75 cm².

Where Sectors of a Circle Show Up

Sectors matter because so many real things are built as slices of a circle, and the area-as-a-fraction idea is exactly what designers need to measure them.

• Windscreen wipers and radar. A wiper or a rotating radar sweep covers a sector; its area is the fraction of the full disc set by the sweep angle, which is how coverage is specified.

• Pie charts and dashboards. Every slice of a pie chart is a sector whose angle is proportional to the data it shows, $90°$ for a quarter, $36°$ for a tenth.

• Pizza, cake, and material cutting. Cutting a round sheet of metal or dough into equal sectors uses the same $\dfrac{360°}{n}$ split; the area of each piece is the circle's area divided by $n$.

• Gears and cams. A cam that engages for part of a rotation acts over a sector of its travel, and engineers size the contact area as a fraction of the full circle.

For a Class 10 student, the sector is where circle geometry meets fractions and angles at once, and mastering "the angle's share of the whole" makes the later mensuration problems, segment areas, and the move into radians feel like one connected idea.

Common Errors When Working With Sectors

Mistake 1: Forgetting the one-half in the arc-length area formula

Where it slips in: A problem gives the arc length and radius but no angle, and the student multiplies them directly.

Don't do this: Write Area $= l \times r$ and stop.

The correct way: The area from the arc length is $\dfrac{1}{2} l,r$. A sector is a wedge filling half its bounding rectangle, so the factor of $\tfrac{1}{2}$ is not optional. The rusher who skips it lands on an answer exactly twice too large.

Mistake 2: Mixing degrees and radians in the same formula

Where it slips in: A student uses the radian formula $\tfrac{1}{2}r^2\theta$ but plugs in the angle in degrees.

Don't do this: Put $\theta = 60$ into $\tfrac{1}{2}r^2\theta$ when 60 means degrees.

The correct way: Match the formula to the unit. Degrees use $\dfrac{\theta}{360°}\times\pi r^2$; radians use $\tfrac{1}{2}r^2\theta$. If the angle is in degrees and you want the radian formula, convert first: radians $= \text{degrees} \times \dfrac{\pi}{180}$.

Mistake 3: Confusing a sector with a segment

Where it slips in: A student computes a sector area when the problem asks for the segment, or the reverse.

Don't do this: Treat the chord-bounded region and the radii-bounded region as the same.

The correct way: A sector is bounded by two radii and an arc; a segment is bounded by a chord and an arc. The segment is the sector with the central triangle removed, so segment area = sector area − triangle area.

Key Takeaways

• A sector of a circle is the region between two radii and an arc; the smaller is the minor sector, the larger the major sector.

• A sector's area is the angle's fraction of the whole circle: $\dfrac{\theta}{360°}\times\pi r^2$ in degrees, or $\tfrac{1}{2}r^2\theta$ in radians.

• The arc length is $\dfrac{\theta}{360°}\times 2\pi r$ (degrees) or $r\theta$ (radians), and the perimeter is $2r + l$.

• With no angle given, the area from the arc length is $\dfrac{1}{2} l,r$ — and the $\tfrac{1}{2}$ is the most-forgotten factor.

• A sector is bounded by two radii; a segment is bounded by a chord, so segment area = sector area − triangle area.

Practice These Problems to Solidify Your Understanding

1. Find the area of a sector of radius 14 cm with a central angle of $90°$ (use $\pi = \tfrac{22}{7}$).

2. A sector has an arc length of 8 cm and a radius of 5 cm. Find its area.

3. Find the perimeter of a sector of radius 9 cm whose arc length is 15 cm.

Answer to Question 1: 154 cm². Answer to Question 2: 20 cm². Answer to Question 3: 33 cm. If Question 2 gave you 40, you skipped the $\tfrac{1}{2}$ in the arc-length area formula (see Mistake 1).

Want a live Bhanzu trainer to walk your child through sectors, segments, and circle mensuration? Book a free demo class — online globally.

Also Read:

• Geometry — Concepts, Formulas, Shapes & Topics Guide

• Arc Length

• Central Angle in Geometry

• Area of a Circle

• Parts of a Circle

• Radius of a Circle

• Diameter of a Circle