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Chapter 1: Problem 66

Express the radius \(r\) of a circle as a function of its area \(A\).

Short Answer

Expert verified
The radius \( r \) is given by: \[ r = \sqrt{ \frac{A}{\pi} } \]

Step by step solution

01

Identify the formula for the area of a circle

The area of a circle is given by the formula: \[ A = \pi r^2 \]
02

Isolate the radius

To express the radius \( r \) as a function of its area \( A \), solve for \( r \) in terms of \( A \). Start by isolating \( r^2 \) on one side of the equation: \[ r^2 = \frac{A}{\pi} \]
03

Solve for the radius \( r \)

Take the square root of both sides to solve for \( r \): \[ r = \sqrt{ \frac{A}{\pi} } \]

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Key Concepts

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

radius of a circle
The radius of a circle is a key measurement in geometry. It is the distance from the center of the circle to any point on its circumference. This distance is always the same, no matter where you measure it along the perimeter of the circle. Knowing the radius helps us find other vital properties of the circle, such as its area and circumference.
circle area formula
The area of a circle is the amount of space enclosed within its boundary. It can be calculated using the formula: This formula allows you to calculate the area if you know the radius. Geometry often requires converting between different properties, and the area formula is one of the most valuable tools for solving circle-related problems.
solving for a variable in a formula
When dealing with formulas, sometimes we need to find the value of one variable in terms of another. To do this, we 'solve for' the desired variable. Here's how we do it with the area of a circle:
  • Start with the area formula:
  • Rearrange the formula to isolate the variable you want to solve for. In this case, we isolate by dividing both sides of the equation by :
  • Finally, solve for by taking the square root of both sides: .

Rearranging equations

This process of isolating the variable and solving for it is key in many areas of mathematics and science. It allows us to understand and predict patterns and relationships.
square root operation
Taking the square root is a mathematical operation that finds a number which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because . In our formula for the radius, we use the square root to solve for . Specifically, we take the square root of to find . The notation tells us to take the square root of the expression inside the parentheses.
geometry
Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space. Circles are one of the fundamental shapes in geometry. Understanding how to calculate the radius, area, and other properties of a circle is essential:
  • It helps in practical applications like engineering and design.
  • It also provides a basis for more advanced topics in mathematics.
  • Geometry skills are crucial for solving real-world problems like finding the shortest distance between points.
With the concepts of radius and area, we can better understand and describe the world around us.

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Most popular questions from this chapter

True or false? Do not use a calculator. $$ \sin \left(91^{\circ}\right)=-\sin \left(89^{\circ}\right) $$

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