Circles: Area - Varsity Tutors

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Circles: Area

Study Guide

Key Definition

The area of a circle is the number of square units that can fit inside that circle. It's given by the formula $A = \pi r^2$ where $r$ is the radius of the circle.

Important Notes

  • The radius of a circle is half its diameter.
  • The constant $\pi$ is approximately equal to 3.14
  • The area of a circle increases quadratically with its radius.
  • Units of area are always square units (such as square inches, square feet, or square meters).

Mathematical Notation

$\pi$ is the mathematical constant approximately equal to 3.14$r$ is the radius of the circle$d$ is the diameter of the circle, $d = 2r$Remember to use proper notation when solving problems

Why It Works

To derive $A = \pi r^2$, imagine dividing the circle into n equal sectors each of central angle 2π/n. Each sector can be viewed as a narrow triangle whose base is the arc length (2πr/n) and whose height is the radius r. If you rearrange these sectors alternately, they form a shape that approaches a parallelogram with base equal to half the circumference (πr) and height r. Taking the limit as n → ∞ makes this exact, giving area = base × height = πr · r = πr².

Remember

Always be careful with units when calculating the area. If the radius is given in feet, the area will be in square feet.

Quick Reference

Area of a circle:$A = \pi r^2$

Understanding Circles: Area

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Beginner Explanation

The area of a circle is the space enclosed by the circle. It is calculated using the formula $A = \pi r^2$ where $r$ is the radius of the circle.Now showing Beginner level explanation.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

What is the area of a circle with radius $4 cm$?

A$8 \pi cm^2$B$16 \pi cm^2$C$4 \pi cm^2$D$2 \pi cm^2$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Pizza Party

Your family is having a pizza party and you're in charge of ordering the pizzas. The pizza place offers two sizes: small with a diameter of 10 inches and large with a diameter of 16 inches. Which size gives more pizza per inch of diameter?Show AnswerClick to reveal the detailed solution for this question exercise.3

Advanced Challenge

Thinking ExerciseAdvanced

Think About This

If the radius of a circle is doubled, what happens to its area?

Show AnswerClick to reveal the detailed explanation for this thinking exercise.

Recap

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