How to Find the Central Angle of a Circle?
TL;DR: Picture a pizza, then draw two cuts from the very center out to the crust. The angle those cuts form is a central angle. Its vertex sits right at the circle's middle. Here's the handy fact you'll lean on constantly: a central angle's degree measure is exactly equal to the arc it carves out on the crust. To turn that arc measure into actual arc length, take the angle as a fraction of 360 and multiply by the circle's full circumference. One simple proportion, no mystery.
Key takeaways:
- A central angle is formed by two radii of the same circle.
- Central angle measure = intercepted arc measure (in degrees).
- Full circle = \(360^\circ\); semicircle = \(180^\circ\); quarter-circle = \(90^\circ\).
- Arc length: \(s = \dfrac{\theta}{360} \times 2\pi r\), or in radians \(s = r\theta\).
- Sector area: \(A = \dfrac{\theta}{360} \times \pi r^2\).
The central angle is useful for dividing a circle into sections. A slice of pizza is a good example of a central angle. A pie chart consists of several sections and helps to display different values.
Type the problems from this lesson and check each step.
A step-by-step guide to finding the central angle of a circle
The central angle is the angle subtended by an arc of a circle at the center of a circle. Radius vectors form the arms of the central angle.
In other words, it is an angle whose vertex is the center of a circle whose arms are two radii lines that intersect at two different points in the circle. When these two points are connected, they form an arc. The central angle is the angle subtended by this arc at the center of the circle.
How to find the central angle of a circle?
The central angle is the angle between the two radii of the circle. To find the central angle, we need to find the length of the arc and the length of the radius. The following steps show how to calculate the central angle in radians.
There are three simple steps to finding the central angle:
- Identify the ends of the arc and the center of the circle (curve). \(AB\) is the arc of the circle and \(O\) is the center of the circle.
- Connect the end of the arc with the center of the circle. Also, measure the length of the arc and its radius. Here \(AB\) is the arc length and \(OA\) and \(OB\) are the radii of the circle.
- Divide the length of the curve by the radius to get the central angle. Using the following formula, we will find the value of the central angle in radians.
\(\color{blue}{Central\:Angle=\frac{Length\:of\:the\:Arc}{Radius}}\)
Finding the Central Angle of a Circle, Example 1:
If the arc length is \(8\) inches and the central angle is \(120\) degrees, find the radius of the arc.
Solution:
\(Radius\:of\:the\:arc= 8\space inches\)
\(Central\:angle= 120°\)
\(Central\:angle\:=\frac{\left(length\:of\:arc\:×\:360°\right)}{\left(2\pi \:×\:radius\right)}\)
\(radius\:=\frac{\left(length\:of\:arc\:×360°\right)}{\left(2\pi \:×\:Central\:angle\right)}\)
\(radius\:=\frac{\left(8\:×\:360°\right)}{\left(2\pi \:×120°\right)}\)
\(radius\:=\frac{12}{\pi \:}\)
Exercises for Finding the Central Angle of a Circle
- From the diagram shown, find the \(m∠BOD\) measure.
- From the diagram shown, find the \(m∠arc MNK\) measure.
- If the arc length measurement is about \(21\space cm\) and the length of the radius measures \(10\space cm\), find the central angle.
- \(\color{blue}{60°}\)
- \(\color{blue}{234°}\)
- \(\color{blue}{120.38°}\)
Recommended EffortlessMath Books
For a deeper walk through every geometry skill from the ground up, Geometry for Beginners covers angles, area, volume, triangles, and transformations with worked examples and plenty of practice. For algebra-heavy geometry topics, the companion Algebra I for Beginners ties the coordinate-plane work back to linear equations.
Frequently Asked Questions
What is the formula to find the central angle of a circle?
The main formula comes from arc length: θ = (s ÷ 2πr) × 360°, where s is the arc length and r is the radius. In radians it is simply θ = s ÷ r. If you know the intercepted arc’s measure in degrees instead, the central angle equals that arc measure, so a 70° arc gives a 70° central angle.
How do you find the central angle of a circle with radius and arc length?
Divide the arc length by the radius to get the angle in radians, then convert if you need degrees. For a circle with radius 5 cm and an arc length of 10 cm, θ = 10 ÷ 5 = 2 radians. Multiply by 180/π to get degrees: 2 × 180/π ≈ 114.6°.
How do you find the central angle of a circle without arc length?
Use whatever else the problem gives you. If you know the sector’s share of the circle, multiply that fraction by 360°, so one-fifth of a circle is 72°. If you know the sector area A and radius r, use θ = (A ÷ πr²) × 360°. And if an inscribed angle intercepts the same arc, the central angle is twice the inscribed angle.
How do you find the central angle of a circle with radius and area?
Use the sector area formula and solve for θ: θ = (A ÷ πr²) × 360°. For a sector with an area of 12π square inches in a circle of radius 6 inches, the whole circle’s area is 36π, so the sector is 12π ÷ 36π = 1/3 of the circle. The central angle is 1/3 × 360° = 120°.
How do you find the central angle of a circle graph?
Multiply each category’s percent by 360°. If 30% of students in a survey chose soccer, that slice has a central angle of 0.30 × 360° = 108°. To check your work, add all the central angles: they should total 360°. If you start with counts instead of percents, divide each count by the total first.
How do you find the radian measure of the central angle of a circle?
Use θ = s ÷ r, where s is the arc length and r is the radius, both in the same units. An arc of 9 m on a circle with a radius of 3 m gives θ = 9 ÷ 3 = 3 radians. If the angle is given in degrees, multiply by π/180 instead: 150° × π/180 = 5π/6 radians.
Related EffortlessMath Lessons
If a topic on this page feels rusty, these short lessons go deeper:
- Types of angles
- How to find arc length
- Complementary, supplementary, and vertical angles
- How to measure angles
- How to find the equation of a circle
Related to This Article
More math articles
- 1994 Realignment: Polarization and Clinton Impeachment
- Mechanical Cleaning: Washers, Ultrasonics, and Cart Washers
- The Best Grade 4 Math Book for Vermont Students
- The demand for a Bill of Rights
- How to Round Decimals? (+FREE Worksheet!)
- The Ultimate 7th Grade SBAC Math Course (+FREE Worksheets)
- Free Grade 3 English Worksheets for Hawai’i Students
- 5th Grade MAAP Math Worksheets: FREE & Printable
- How to Ace AP Calculus AB in 2027: Self-Study or Classroom Playbook
- How to Calculate Segment Lengths in Circles Using the Power Theorem
What people say about "How to Find the Central Angle of a Circle"?
No one replied yet.