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Chapter 9: Problem 63

The area \(\mathscr{A}\) of a circle with radius \(r\) is given by the formula $$\mathscr{A}=\pi r^{2}$$ If a circle has area \(81 \pi\) in. , what is its radius?

Short Answer

Expert verified
The radius is 9 inches.

Step by step solution

01

Write down the formula for the area of a circle

The formula for the area of a circle is given by \( \mathscr{A} = \pi r^{2} \).
02

Substitute the given area into the formula

We are given that the area \( \mathscr{A}\) is \( 81 \pi \) square inches. Substitute this value into the formula: \( 81 \pi = \pi r^{2} \).
03

Solve for the radius

Divide both sides of the equation by \( \pi \) to isolate \( r^{2} \): \[ 81 \pi = \pi r^{2} \]\[ 81 = r^{2} \].
04

Take the square root

Take the square root of both sides of the equation to solve for \( r \): \[ r = \sqrt{81} \] \[ r = 9 \].

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Key Concepts

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

radius
The radius of a circle is the line segment from the center of the circle to any point on its circumference. It is an essential characteristic of a circle because it defines its size.
The radius is usually denoted by the letter 'r'. When dealing with circle-related problems, you will often need to use the radius to find other properties of the circle like its area or circumference.
For example, if you know the area of a circle, you can reverse-calculate the radius using the circle area formula.
area of a circle
The area of a circle represents the region occupied by the circle in a two-dimensional plane. The formula to calculate the area of a circle is often given as: \(\text{Area} = \pi r^{2}\)
Here, \(\text{Area}\) denotes the area of the circle, \(\pi\) is a constant approximately equal to 3.14159, and \(r\) is the radius of the circle. This formula is key when you need to find out how much space a circle covers.
For instance, if the area of a circle is given as \(81 \pi\) square inches, you can plug this value into the formula to find the radius.
square root
The square root is a mathematical function that determines what number, when multiplied by itself, gives the original number. It is represented by the symbol \(\sqrt{}\).
For example, the square root of \(81\) is \(9\) because \(9 \times 9 = 81\).
In the context of the circle area problem, once we isolate \(r^{2}\), which equals 81, finding the radius \(r\) involves taking the square root of 81 to get \(r = 9\).
This step is essential in solving many geometric and algebraic problems.

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Most popular questions from this chapter

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When a student was asked to solve \(x^{2}=81,\) she wrote \(\\{9\\}\) as her answer. Her teacher did not give her full credit. The student argued that because \(9^{2}=81,\) her answer had to be correct. WHAT WENT WRONG? Give the correct solution set.

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