Table of Contents
Understanding Arcs
In a circle, an arc is a continuous part of the circumference between two points on the circle.
The two points that mark the ends of an arc are called its endpoints. The center of the circle and these two endpoints form a central angle, and that central angle is closely related to the size of the arc.
There are several common types of arcs:
- Minor arc: The “shorter way around” between two points on a circle. Its measure is less than $180^\circ$.
- Major arc: The “longer way around” between the same two points. Its measure is greater than $180^\circ$.
- Semicircle: An arc whose endpoints are the endpoints of a diameter; its measure is exactly $180^\circ$.
By convention:
- A minor arc is usually named with two letters, like $\widehat{AB}$.
- A major arc or semicircle is often named with three letters to avoid confusion, like $\widehat{ACB}$, where $C$ is a point on the arc between $A$ and $B$.
Measuring Arcs
Degree measure of an arc
The degree measure of a minor arc is equal to the measure of its central angle.
If central angle $\angle AOB$ (with vertex at the center $O$) measures $\theta^\circ$, then the measure of the arc $\widehat{AB}$ is also $\theta^\circ$.
For a full circle:
- The central angle is $360^\circ$.
- The arc measure of the entire circle is $360^\circ$.
A major arc has measure:
$$
m(\text{major arc } A\text{–}B) = 360^\circ - m(\text{minor arc } A\text{–}B).
$$
Radian measure of an arc
An arc can also be measured in radians. For a central angle $\theta$ measured in radians, the corresponding arc has radian measure $\theta$ as well. A full circle is $2\pi$ radians.
The connection between degrees and radians:
- $360^\circ = 2\pi$ radians
- $180^\circ = \pi$ radians
Conversions:
- Degrees to radians: $\theta_{\text{rad}} = \theta_{\text{deg}} \cdot \dfrac{\pi}{180}$
- Radians to degrees: $\theta_{\text{deg}} = \theta_{\text{rad}} \cdot \dfrac{180}{\pi}$
Arc Length
Arc length is the actual distance along the curved part of the circle between two points, just like measuring the length of a piece of string laid along the arc.
Let:
- $r$ be the radius of the circle,
- $\theta$ be the measure of the central angle that intercepts the arc.
There are two standard formulas for arc length.
Using radians
When $\theta$ is in radians:
$$
\text{Arc length} = s = r\theta.
$$
This is the most direct formula. For example, if $r = 5$ and $\theta = \dfrac{\pi}{3}$,
$$
s = 5 \cdot \dfrac{\pi}{3} = \dfrac{5\pi}{3}.
$$
Using degrees
When $\theta$ is in degrees, the arc is a fraction of the full circle. The full circumference is $2\pi r$, and the fraction is $\dfrac{\theta}{360}$, so:
$$
s = \frac{\theta}{360^\circ}\cdot 2\pi r.
$$
This expresses the idea:
Arc length = (fraction of full circle) × (circumference).
For example, for a $60^\circ$ arc on a circle of radius $10$:
$$
s = \frac{60}{360}\cdot 2\pi\cdot 10 = \frac{1}{6}\cdot 20\pi = \frac{20\pi}{6} = \frac{10\pi}{3}.
$$
Understanding Sectors
A sector of a circle is the region inside the circle bounded by:
- two radii, and
- the arc between their endpoints.
It looks like a “slice of pizza” or “wedge” of the circle.
Like arcs, sectors can be:
- Minor sector: formed by a minor arc and its radii.
- Major sector: formed by a major arc and its radii.
The size of a sector is determined by the measure of its central angle.
Area of a Sector
A full circle of radius $r$ has area:
$$
A_{\text{circle}} = \pi r^2.
$$
A sector represents a fraction of the full circle, determined by the central angle.
Let:
- $r$ be the radius,
- $\theta$ be the central angle.
Again, we have two standard formulas depending on the unit for $\theta$.
Using degrees
If $\theta$ is measured in degrees:
$$
\text{Area of sector} = A = \frac{\theta}{360^\circ}\cdot \pi r^2.
$$
This uses the idea:
Area of sector = (fraction of full circle) × (area of full circle).
Example: If $r = 6$ and $\theta = 90^\circ$:
$$
A = \frac{90}{360}\cdot \pi \cdot 6^2
= \frac{1}{4}\cdot \pi\cdot 36
= 9\pi.
$$
Using radians
If $\theta$ is measured in radians:
$$
A = \frac{1}{2}r^2\theta.
$$
This formula is especially convenient when angles are naturally given in radians.
Example: If $r = 4$ and $\theta = \dfrac{\pi}{2}$,
$$
A = \frac{1}{2}\cdot 4^2\cdot \frac{\pi}{2}
= \frac{1}{2}\cdot 16\cdot \frac{\pi}{2}
= 4\pi.
$$
Relationships Between Arc Length and Sector Area
For a given circle with radius $r$ and central angle $\theta$ (in radians):
- Arc length:
$$
s = r\theta
$$ - Sector area:
$$
A = \frac{1}{2}r^2\theta
$$
We can relate $A$ and $s$ directly by eliminating $\theta$.
From $s = r\theta$ we get $\theta = \dfrac{s}{r}$. Substitute into the area formula:
$$
A = \frac{1}{2}r^2\cdot \frac{s}{r} = \frac{1}{2}rs.
$$
So for a given circle, another useful formula is:
$$
A = \frac{1}{2} \, r \, s,
$$
where:
- $A$ is the area of the sector,
- $r$ is the radius,
- $s$ is the length of the arc bounding the sector.
This formula can be helpful when you know the radius and arc length but not the angle.
Working With Fractions of a Circle
Many arc and sector problems use common fractions of a circle. These often correspond to familiar angles:
- Quarter circle: $\dfrac{1}{4}$ of the circle, $\theta = 90^\circ = \dfrac{\pi}{2}$.
- Half circle: $\dfrac{1}{2}$ of the circle, $\theta = 180^\circ = \pi$.
- Third of a circle: $\dfrac{1}{3}$ of the circle, $\theta = 120^\circ = \dfrac{2\pi}{3}$.
- Sixth of a circle: $\dfrac{1}{6}$ of the circle, $\theta = 60^\circ = \dfrac{\pi}{3}$.
For any such fraction $f$ of a circle:
- Arc length: $s = f \cdot 2\pi r$
- Sector area: $A = f \cdot \pi r^2$
Typical Types of Questions
In this chapter, the main kinds of tasks involve using the formulas in different directions:
- Find arc length when given radius and angle.
- Find sector area when given radius and angle.
- Find the angle (in degrees or radians) when given arc length and radius.
- Find the radius when given arc length and angle, or when given sector area and angle.
- Convert between degrees and radians to use the appropriate formula.
These problems often require:
- Choosing whether to work in degrees or radians,
- Interpreting “fraction of a circle” language,
- Knowing which formula matches the given data.