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Chapter 0: Problem 15

Find the area of a circle with radius 3 in.

Short Answer

Expert verified
The area is \( 9\pi \) square inches.

Step by step solution

01

Recall the formula for the area of a circle

The area of a circle is given by the formula \( A = \pi r^2 \), where \( r \) is the radius of the circle.
02

Substitute the radius into the formula

Given the radius \( r = 3 \) inches, substitute \( r \) into the formula: \( A = \pi (3)^2 \).
03

Perform the calculation

Calculate the square of 3: \( 3^2 = 9 \). Then the area is \( A = 9\pi \) square inches.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radius
The radius is a fundamental concept in geometry, relating to circles. It is the distance from the center of a circle to any point on its boundary. In the given exercise, the radius is 3 inches.
The radius is crucial in many calculations, including finding the area and circumference of circles.
  • Symbol: Often represented by 'r'.
  • Measuring units: The radius can be given in any unit of length, such as inches, centimeters, or meters.
  • Example: If you draw a line from the center of a circle to its edge, the length of that line is the radius.
Understanding the radius helps in utilizing other geometric formulas effectively.
Pi
Pi, represented by the symbol \(\pi\), is a crucial mathematical constant in geometry and trigonometry. It represents the ratio of the circumference of a circle to its diameter. Pi is approximately equal to 3.14159, but it is an irrational number, meaning it has an infinite number of non-repeating decimals.
In calculations involving circles, Pi is indispensable:
  • In the formula for the area of a circle \(A = \pi r^2\), Pi connects the radius to the area.
  • It also appears in the formula for the circumference of a circle, \(\pi d\), where 'd' is the diameter.
Memorizing the value of Pi and its significance will make solving circle-related problems easier.
Geometry
Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space. The study of circles is a vital part of geometry, covering concepts such as radius, diameter, circumference, and area.
In solving the area of a circle, as in the given exercise, you use geometric principles and formulas. The formula \(A = \pi r^2\) is a prime example of how geometry simplifies complex measurements:
  • The radius (r) helps determine the size of the circle.
  • Pi (\(\pi\)) provides a constant that simplifies the relationship between the radius and area.
  • Understanding these relationships and formulas is key to mastering geometry.
Geometry not only deals with circles but also with various other shapes and forms, constructing a foundation for many real-life applications.

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Most popular questions from this chapter

A side of one rectangle is \(25 \mathrm{ft}\) and the corresponding side of a second similar rectangle is \(10 \mathrm{ft}\). If the perimeter of the first rectangle is \(150 \mathrm{ft}\) and its area is 1250 sq ft, find the perimeter and area of the similar rectangle.

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