Arc Lengths and Areas of Circular Sectors and Segments

Hello again. In the previous lesson, tangent questions became manageable once you marked the guaranteed facts: a radius meets a tangent at $90^{\circ }$, and tangent segments from the same external point are equal. This lesson keeps the same diagram-first approach, but now the focus is on parts inside or along the edge of a circle.

You will calculate:

• the length of an arc;
• the area of a sector;
• the area of a segment, using the sector minus the triangle inside it.

The central idea is simple: an angle at the centre tells you what fraction of the full circle you are dealing with.

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1. Identify the part of the circle before choosing a formula

A full circle has:

$$360^{\circ }$$

So a central angle of $90^{\circ }$ describes one quarter of a circle, while $180^{\circ }$ describes one half.

Three names matter:

• An arc is a curved part of the circle’s circumference.
• A sector is the wedge-shaped region bounded by two radii and an arc.
• A segment is the region bounded by an arc and a chord. A chord is a straight line joining two points on the circle.

A useful first step in every question is to shade or trace the requested region. This prevents a frequent mistake: calculating the area of a sector when the question actually asks for the area of the smaller curved segment.

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2. The “fraction of a circle” principle

Let the central angle be $\theta$. The sector is:

$$\frac{\theta }{360}$$

of the entire circle.

The area of a full circle is:

$$\pi r^2$$

and its circumference is:

$$2\pi r$$

Therefore, multiply the relevant whole-circle formula by $\frac{\theta }{360}$.

Quantity required	Formula	Units
Arc length	$\frac{\theta }{360}\times 2\pi r$	length units, such as $\text{cm}$
Area of a sector	$\frac{\theta }{360}\times \pi r^2$	square units, such as ${\text{cm}}^2$

Notice the difference:

• arc length is part of the circumference, so it uses $2\pi r$;
• sector area is part of the circle’s area, so it uses $\pi r^2$.

Worked example: an arc and sector from the same diagram

A circle has radius $7\text{ cm}$, and a sector has central angle $140^{\circ }$.

First find the fraction of the circle:

$$\frac{140}{360}=\frac{7}{18}$$

Arc length

$$\text{arc length}=\frac{140}{360}\times 2\pi (7)$$ $$=\frac{7}{18}\times 14\pi$$ $$=\frac{49\pi }{9}$$ $$\approx 17.1\text{ cm}$$

Area of the sector

$$\text{area of sector}=\frac{140}{360}\times \pi (7{)}^2$$ $$=\frac{7}{18}\times 49\pi$$ $$=\frac{343\pi }{18}$$ $$\approx 59.9{\text{ cm}}^2$$

The same fraction, $\frac{140}{360}$, is used both times. What changes is the full-circle quantity being shared out: circumference for an arc, area for a sector.

Sector perimeter

Occasionally the question asks for the perimeter of a sector. Its boundary contains two radii and one arc:

$$\text{perimeter of sector}=2r+\text{arc length}$$

For the $140^{\circ }$, $r=7\text{ cm}$ example:

$$\text{perimeter}=14+\frac{49\pi }{9}$$ $$\approx 31.1\text{ cm}$$

Do not confuse sector perimeter with sector area: perimeter has units of $\text{cm}$, while area has units of ${\text{cm}}^2$.

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3. Minor and major arcs or sectors

A minor arc or sector is the smaller one, with angle less than $180^{\circ }$. A major arc or sector is the larger one, with angle greater than $180^{\circ }$.

If a diagram gives a small central angle but asks for the major arc or sector, calculate the missing central angle first:

$$\text{major angle}=360^{\circ }-\text{minor angle}$$

For example, if the minor angle is $50^{\circ }$, then the major angle is:

$$360^{\circ }-50^{\circ }=310^{\circ }$$

Suppose the radius is $6\text{ cm}$, and you need the major arc length.

$$\text{major arc length}=\frac{310}{360}\times 2\pi (6)$$ $$=\frac{31\pi }{3}$$ $$\approx 32.5\text{ cm}$$

A quick reasonableness check helps. The entire circumference is:

$$2\pi (6)=12\pi \approx 37.7\text{ cm}$$

A major arc should be most of that circumference, so $32.5\text{ cm}$ is sensible. An answer near $5.2\text{ cm}$ would describe the minor arc instead.

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4. Area of a segment: sector minus triangle

A segment is not a wedge. It is the curved “cap” between a chord and an arc.

The segment sits inside a sector, but the triangle formed by the two radii is not part of the segment. Therefore:

$$\text{area of segment}=\text{area of sector}-\text{area of triangle}$$

Worked example: a $90^{\circ }$ segment

A circle has radius $8\text{ cm}$. A chord joins the endpoints of two radii with a central angle of $90^{\circ }$. Find the area of the minor segment.

First, calculate the sector area:

$$\text{area of sector}=\frac{90}{360}\times \pi (8{)}^2$$ $$=\frac{1}{4}\times 64\pi$$ $$=16\pi {\text{ cm}}^2$$

Now calculate the triangle area. Since the central angle is $90^{\circ }$, the two radii form perpendicular sides of length $8\text{ cm}$:

$$\text{area of triangle}=\frac{1}{2}\times 8\times 8$$ $$=32{\text{ cm}}^2$$

Finally subtract:

$$\text{area of segment}=16\pi -32$$ $$\approx 18.3{\text{ cm}}^2$$

Keep $\pi$ in the calculation until the final line unless the question tells you otherwise. Rounding early can make the final answer less accurate.

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5. Finding the triangle area in a segment question

The triangle inside a segment may not always be right-angled. Look at the information supplied and select an appropriate triangle-area method.

When base and perpendicular height are given

Use:

$$\text{area of triangle}=\frac{1}{2}\times \text{base}\times \text{perpendicular height}$$

For example, suppose a segment has:

• radius $4\text{ cm}$,
• central angle $160^{\circ }$,
• chord length $6\text{ cm}$,
• perpendicular distance from the centre to the chord $1.5\text{ cm}$.

The sector area is:

$$\frac{160}{360}\times \pi (4{)}^2=\frac{64\pi }{9}$$

The triangle’s base is the chord, and the perpendicular height is $1.5\text{ cm}$:

$$\text{area of triangle}=\frac{1}{2}\times 6\times 1.5$$ $$=4.5{\text{ cm}}^2$$

So the segment area is:

$$\frac{64\pi }{9}-4.5$$ $$\approx 17.8{\text{ cm}}^2$$

When two radii and the included angle are given

If the radius is $r$ and the central angle is $\theta$, the triangle has two sides of length $r$, with included angle $\theta$. Its area can be found using:

$$\text{area of triangle}=\frac{1}{2}{r}^2\sin \theta$$

Make sure your calculator is in degree mode when using an angle measured in degrees.

For a circle of radius $10\text{ cm}$ with central angle $60^{\circ }$:

$$\text{area of sector}=\frac{60}{360}\times \pi (10{)}^2$$ $$=\frac{50\pi }{3}$$

The triangle area is:

$$\frac{1}{2}(10{)}^2\sin 60^{\circ }$$ $$=50\times \frac{\sqrt{3}}{2}$$ $$=25\sqrt{3}$$

Therefore the segment area is:

$$\frac{50\pi }{3}-25\sqrt{3}$$ $$\approx 9.1{\text{ cm}}^2$$

The subtraction makes geometric sense: a $60^{\circ }$ sector is fairly narrow, and the triangle occupies most of that wedge, leaving a relatively small curved cap.

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6. Major segments and a dependable solving routine

A chord divides a circle into a minor segment and a major segment. If you have found the minor segment and need the major segment, use the whole circle:

$$\text{area of major segment}=\pi r^2-\text{area of minor segment}$$

This is usually easier and safer than trying to work directly with the larger shape.

For every arc, sector, or segment question, use this routine:

1. Mark the central angle that belongs to the requested region.
   If the question asks for a major region but shows a minor angle, use $360^{\circ }-\theta$.

2. Decide whether the answer is a length or an area.
   Arc length uses circumference; sector and segment questions use areas.

3. For a segment, split the work into two areas.
   Find the sector area and triangle area separately before subtracting.

4. Check units.
   Arc lengths and perimeters use ordinary units. Areas use squared units.

5. Check the size of the answer.
   A minor arc must be shorter than half the circumference, and a minor segment must be smaller than its sector.

Common errors to avoid

Error	Better habit
Using $\pi r^2$ for arc length	Arc length is part of the circumference, so start with $2\pi r$.
Forgetting that a given angle is minor when the major region is requested	Find the major angle using $360^{\circ }-\theta$.
Calling a sector a segment	A sector has two radii; a segment has a chord.
Stopping after finding sector area in a segment question	Subtract the triangle area.
Giving $\text{cm}$ for area	Use ${\text{cm}}^2$, ${\text{m}}^2$, or the relevant squared unit.
Rounding every intermediate result	Keep exact values or calculator digits until the final answer.

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Key takeaways

Every calculation in this lesson starts from the same fraction:

$$\frac{\theta }{360}$$

Use it with the circumference for an arc:

$$\text{arc length}=\frac{\theta }{360}\times 2\pi r$$

Use it with the circle’s area for a sector:

$$\text{area of sector}=\frac{\theta }{360}\times \pi r^2$$

For a segment, first identify the sector containing it, then subtract the triangle:

$$\text{area of segment}=\text{area of sector}-\text{area of triangle}$$

Finally, always check whether the question asks for a minor or major region and make your units match what you calculated.

You have now completed the circle-measurement work in this revision module. Next, the course returns to algebra: constructing a quadratic polynomial when its zeroes or other conditions are given.