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Arcs

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Arcs

Study Guide

Key Definition

An arc is a portion of the circumference of a circle. A minor arc is shorter than a semicircle, and a major arc is longer. The measurement of an arc is equivalent to the central angle that intercepts it.

Important Notes

The measure of a minor arc is less than $180^\circ$.
The measure of a major arc is greater than $180^\circ$.
An arc measuring exactly $180^\circ$ is called a semicircle.
The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

Mathematical Notation

$m\widehat{AB} = x^\circ$ means the measure of arc $\widehat{AB}$ is $x^\circ$.

Remember to use proper notation when solving problems

Why It Works

Arcs work because they are portions of the circumference of a circle. They are measured in degrees, just like angles, which makes them easy to work with in geometry.

Remember

Remember, any arc on a circle is defined by its two endpoints.

Quick Reference

Minor Arc:$< 180^\circ$

Major Arc:$> 180^\circ$

Arc Addition Postulate:$m\widehat{PQR} = m\widehat{PQ} + m\widehat{QR}$

Understanding Arcs

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

An arc is a portion of the circumference of a circle, determined by two endpoints on the circle.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If the measure of angle $\angle AOB$ is $90^\circ$, what is the measure of arc $\widehat{AB}$?

Real-World Problem

Question Exercise

Intermediate

Pizza Party Scenario

You are at a pizza party. You notice that one slice of pizza forms an angle of $30^\circ$ at the center of the pizza. What is the measure of the arc that the slice of pizza intercepts?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

What happens to the measure of an arc if you double the measure of the central angle that intercepts it?

Challenge Quiz

Single Choice Quiz

Advanced

In a circle where A and B are the only marked points on the circumference, if the measure of arc $\widehat{AB}$ is $100^\circ$, what is the measure of the other (major) arc determined by A and B?

Recap

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Review key concepts and takeaways