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Sectors

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Sectors

Study Guide

Key Definition

A sector is a region of a circle bounded by a central angle and its intercepted arc. The area of a sector is calculated using the formula $\frac{\text{central angle}}{360^\circ} \times \pi r^2$.

Important Notes

A sector can be thought of as a 'slice' of a circle.
The central angle is measured in degrees.
The formula $\frac{\text{central angle}}{360^\circ} \times \pi r^2$ gives the area of the sector.
If the central angle is $360^\circ$, the sector is the entire circle.
The intercepted arc is the arc enclosed by the sides of the angle.

Mathematical Notation

$\pi$ is approximately 3.14159

$\frac{a}{b}$ represents division of a by b

$x^2$ denotes x squared

$360^\circ$ represents a full circle

$\sqrt{x}$ denotes the square root of x

Remember to use proper notation when solving problems

Why It Works

The formula for the area of a sector is derived from the fact that the ratio of the sector's area to the circle's area is equal to the ratio of the central angle to $360^\circ$.

Remember

The key formula for finding the area of a sector is $\frac{\text{central angle}}{360^\circ} \times \pi r^2$.

Quick Reference

Sector Area Formula:$\frac{\text{central angle}}{360^\circ} \times \pi r^2$

Understanding Sectors

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A sector is like a pizza slice from a circle. Its area can be found using $\frac{\text{central angle}}{360^\circ} \times \pi r^2$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the area of a sector with a central angle of $60^\circ$ and a radius of 4 units?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are cutting a pizza into 8 equal slices. Each slice is a sector with a central angle of $45^\circ$. If the pizza has a radius of 12 inches, what is the area of one slice?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Given a circle with radius $r$ and a sector with angle \theta, express the area of the sector in terms of $r$ and $\theta$.

Challenge Quiz

Single Choice Quiz

Advanced

A circle with a radius of 10 units has a sector with a central angle of $120^\circ$. What is the area of this sector?

Recap

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Review key concepts and takeaways