Arcs and Sectors Using Degrees
Introduction
students, imagine a pizza cut into slices 🍕. If you know the size of the whole pizza and the angle of one slice, you can find the length of the crust on that slice and the area of the slice itself. In geometry, that slice is called a sector, and the curved edge is called an arc. This lesson explains how to work with arcs and sectors when angles are measured in degrees.
Objectives
By the end of this lesson, students, you should be able to:
- explain the meaning of an arc, a sector, and a central angle,
- use the degree-based formulas for arc length and sector area,
- solve geometry problems involving circles in real-life settings,
- connect these ideas to the wider IB Mathematics Analysis and Approaches SL topic of geometry and trigonometry,
- use examples and calculations to show understanding.
These ideas matter because circles appear everywhere: wheels, clocks, sports tracks, gears, cakes, and even radar screens ⏱️. Learning how arcs and sectors work helps you describe parts of a circle precisely and solve practical problems.
Key ideas and terminology
A circle is the set of all points in a plane that are the same distance from a fixed point called the centre.
A radius is a line segment from the centre to the circle. If the radius is $r$, then the full distance across the circle is the diameter, which is $2r$.
An arc is a part of the circumference of a circle. If you imagine “cutting out” a piece of the edge of a circular pizza, that curved piece is an arc.
A sector is the region enclosed by two radii and the arc between them. It looks like a slice of pie 🥧.
The angle at the centre of the circle is called the central angle. In this lesson, that angle is measured in degrees.
When the central angle is $\theta$ degrees, the arc and sector are proportional to the whole circle. This is the main idea behind all the formulas.
Arc length using degrees
The circumference of a full circle is $2\pi r$. Since a complete circle has $360^\circ$, an arc subtending an angle of $\theta^\circ$ is the fraction $\frac{\theta}{360}$ of the whole circumference.
So the arc length $s$ is
$$s=\frac{\theta}{360}\cdot 2\pi r$$
This formula works because the arc is just a part of the full circle.
Example 1
Find the arc length of a circle with radius $10$ cm and central angle $72^\circ$.
Use
$$s=\frac{\theta}{360}\cdot 2\pi r$$
Substitute $\theta=72$ and $r=10$:
$$s=\frac{72}{360}\cdot 2\pi(10)$$
$$s=\frac{1}{5}\cdot 20\pi$$
$$s=4\pi$$
So the arc length is $4\pi$ cm, which is about $12.6$ cm.
This means the curved part of the circle is about the same length as a large ruler placed end to end with a little extra 📏.
Example 2
A bicycle wheel has radius $35$ cm. What is the arc length of the part of the wheel rotated through $15^\circ$?
$$s=\frac{15}{360}\cdot 2\pi(35)$$
$$s=\frac{15}{360}\cdot 70\pi$$
$$s=\frac{7}{24}\pi$$
So the arc length is $\frac{7\pi}{24}$ cm, which is about $0.92$ cm.
Common mistake
A common error is using $\theta$ directly instead of the fraction $\frac{\theta}{360}$. Remember, the angle must be compared with the full turn of $360^\circ$.
Sector area using degrees
The area of a full circle is $\pi r^2$. A sector is a fraction of the whole circle, so its area is based on the same idea as arc length.
If the central angle is $\theta^\circ$, the sector area $A$ is
$$A=\frac{\theta}{360}\cdot \pi r^2$$
Example 3
Find the area of a sector with radius $8$ cm and angle $45^\circ$.
$$A=\frac{45}{360}\cdot \pi(8)^2$$
$$A=\frac{1}{8}\cdot 64\pi$$
$$A=8\pi$$
So the sector area is $8\pi$ cm^2, or about $25.1$ cm^2.
Example 4
A round cake with radius $12$ cm is cut into a sector with angle $150^\circ$. Find the area of the slice.
$$A=\frac{150}{360}\cdot \pi(12)^2$$
$$A=\frac{5}{12}\cdot 144\pi$$
$$A=60\pi$$
So the slice has area $60\pi$ cm^2, which is about $188.5$ cm^2 🎂.
Why sector area matters
Sector area is useful in situations where a curved region represents part of a larger circular object. For example:
- the swept area of a rotating arm,
- the slice of a pie chart,
- the shape covered by a lighthouse beam,
- the region of a circular garden path.
Working with unknown radius or angle
Sometimes you are not given all the information at once. You may need to rearrange the formulas.
From
$$s=\frac{\theta}{360}\cdot 2\pi r$$
you can solve for $\theta$:
$$\theta=\frac{360s}{2\pi r}$$
or simplify to
$$\theta=\frac{180s}{\pi r}$$
You can also solve for $r$ if needed:
$$r=\frac{360s}{2\pi\theta}$$
Similarly, from
$$A=\frac{\theta}{360}\cdot \pi r^2$$
you can solve for $\theta$ or $r$ depending on the question.
Example 5
A sector has area $30\pi$ cm^2, radius $15$ cm, and angle $\theta$. Find $\theta$.
Start with
$$30\pi=\frac{\theta}{360}\cdot \pi(15)^2$$
$$30\pi=\frac{\theta}{360}\cdot 225\pi$$
Cancel $\pi$:
$$30=\frac{225\theta}{360}$$
Multiply both sides by $360$:
$$10800=225\theta$$
Divide by $225$:
$$\theta=48$$
So the angle is $48^\circ$.
This kind of question is common in IB exams because it tests algebra, circle geometry, and careful reasoning all at once.
Connecting arcs and sectors to the rest of geometry and trigonometry
Arcs and sectors are not isolated facts. They connect to many other parts of mathematics.
Geometry links
- The radius, diameter, circumference, and area of a circle are all related.
- If two circles are similar, their arc lengths and sector areas scale with the radius.
- Problems may combine circles with triangles, especially when radii form isosceles triangles.
For example, if two radii and the chord between them form a triangle, you may use geometry or trigonometry to find missing lengths or angles.
Trigonometry links
Although this lesson uses degrees in circle formulas, trigonometry often appears when you work with triangles inside circles.
For instance, if a radius and chord make an angle, you may need sine, cosine, or tangent to find unknown sides. Also, sector problems can lead to equations involving triangles, where trigonometric reasoning helps solve the full problem.
Real-world connection
A wind turbine blade rotates through an angle, creating a sector-like swept region. A satellite dish uses circular geometry to aim signals. A clock hand moves around a circle, and the path it sweeps is an arc. In each case, $\theta$, $r$, $s$, and $A$ help describe the motion or region precisely 🛰️.
Problem-solving strategy
When students faces an arcs or sectors question, follow these steps:
- Identify the given values: radius $r$, angle $\theta$, arc length $s$, or sector area $A$.
- Decide whether the problem is about length or area.
- Choose the correct formula:
$$s=\frac{\theta}{360}\cdot 2\pi r$$
or
$$A=\frac{\theta}{360}\cdot \pi r^2$$
- Substitute values carefully.
- Keep the exact answer in terms of $\pi$ unless the question asks for a decimal.
- Check that the answer makes sense. A larger angle should give a longer arc and a larger sector area.
Example 6: mixed reasoning
A sector has radius $9$ cm and angle $80^\circ$.
Find the arc length:
$$s=\frac{80}{360}\cdot 2\pi(9)$$
$$s=\frac{2}{9}\cdot 18\pi$$
$$s=4\pi$$
Find the area:
$$A=\frac{80}{360}\cdot \pi(9)^2$$
$$A=\frac{2}{9}\cdot 81\pi$$
$$A=18\pi$$
Notice how the same angle produces both a curve length and a region area, but the formulas are different.
Conclusion
Arcs and sectors using degrees are essential tools in circle geometry. The main idea is proportionality: a sector or arc is a fraction of a full circle based on the central angle $\theta$ out of $360^\circ$. That leads to the key formulas for arc length and sector area. These formulas are useful in real-world situations and also connect to more advanced geometry and trigonometry topics in IB Mathematics Analysis and Approaches SL.
If you can identify the radius, angle, arc length, or sector area, students, you can usually solve the problem by choosing the right formula and working carefully step by step ✅.
Study Notes
- An arc is part of the circumference of a circle.
- A sector is a region bounded by two radii and an arc.
- The central angle is measured at the centre of the circle.
- For degrees, arc length is
$$s=\frac{\theta}{360}\cdot 2\pi r$$
- For degrees, sector area is
$$A=\frac{\theta}{360}\cdot \pi r^2$$
- Use exact values with $\pi$ when possible.
- A full circle is $360^\circ$.
- A larger angle gives a longer arc and a larger sector area.
- Arc and sector problems connect circle geometry with algebra and trigonometry.
- Real-world examples include pizza slices, clock hands, wheels, and rotating machinery.