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Finding Arc Lengths

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Finding Arc Lengths

Study Guide

Key Definition

An arc is a section of a circle's circumference. To find an arc length, use the formula: $\frac{\theta}{360^\circ} \times 2\pi r$, where $\theta$ is the central angle in degrees and $r$ is the radius.

Important Notes

Arc length is a portion of the circumference.
The formula $\frac{\theta}{360^\circ} \times 2\pi r$ helps find the arc length.
Use $\pi \approx 3.14159$ in calculations.
Ensure $\theta$ is in degrees.
Arc length and central angle are directly proportional.

Mathematical Notation

$\theta$ represents the central angle

$r$ is the radius

$\pi$ is approximately 3.14159

$360^\circ$ is a full circle

Remember to use proper notation when solving problems

Why It Works

Since arcs are parts of the circle’s perimeter, the fraction $\frac{\theta}{360^\circ}$ of the full circumference $2\pi r$ gives the arc length.

Remember

Always check the unit of measure for angles and convert if necessary using $\theta\text{ (radians)} = \theta^\circ \times \frac{\pi}{180}$ or $\theta^\circ = \theta\text{ (radians)} \times \frac{180}{\pi}$.

Quick Reference

Arc Length Formula:$\frac{\theta}{360^\circ} \times 2\pi r$

Understanding Finding Arc Lengths

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

An arc length is like the 'crust' of a pizza slice. If you know the angle of the slice and the radius, you can find the arc length using $\frac{\theta}{360^\circ} \times 2\pi r$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If a circle has a radius of $5$ and a central angle of $90^\circ$, what is the arc length?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you’re designing a new pizza with a slice angle of $60^\circ$ and a radius of $10$ cm. Find the arc length of the slice.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Given a circle with radius $7$ meters and an arc with a central angle of $135^\circ$, determine the arc length.

Challenge Quiz

Single Choice Quiz

Advanced

Calculate the arc length for a circle with radius $8$ and a central angle of $225^\circ$.

Recap

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Review key concepts and takeaways