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Chapter 3: Problem 56

Solve each formula for the indicated variable. $$C=2 \pi r\( for \)r$ (geometry)$$

Short Answer

Expert verified
The short answer is: To solve the formula \(C = 2\pi r\) for \(r\), isolate the variable \(r\) by dividing both sides of the equation by \(2\pi\). The final solution is \(r = \frac{C}{2\pi}\).

Step by step solution

01

Understand the given formula

The given formula is \(C = 2\pi r\), which represents the circumference(\(C\)) of a circle with radius(\(r\)). To solve for the radius, we need to isolate \(r\) on one side of the equation.
02

Divide by \(2\pi\)

To isolate \(r\), we need to divide both sides of the equation by \(2\pi\). This will result in the following equation: \[ \frac{C}{2\pi} = \frac{2\pi r}{2\pi} \]
03

Simplify the equation

Now, we can simplify the equation by canceling out the \(\pi\) terms and the 2's on both sides. This leaves us with the following equation: \[ \frac{C}{2\pi} = r \]
04

Final solution

We have successfully solved the equation for the radius (\(r\)). The final solution is: \[ r = \frac{C}{2\pi} \] Now, given any circumference (\(C\)), we can easily find the radius (\(r\)) of the circle by using this equation.

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Key Concepts

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

Geometry
Geometry is the field of mathematics that deals with shapes, sizes, and properties of space. It is used to understand various forms, like circles, triangles, and polygons.
In the context of the exercise we're looking at, geometry specifically deals with circles and their properties.
A circle is a two-dimensional shape where all points are equidistant from a central point, known as the center. This central point is crucial in defining other characteristics of the circle, like its radius and circumference. Circles are one of the primary shapes studied in geometry because of their symmetry and unique properties.
Equations
Equations are mathematical statements that express the equality of two expressions. They are used to solve problems by showing how different quantities relate to each other.
In our exercise, the equation \( C = 2\pi r \) represents the circumference of a circle where \( C \) is the circumference, \( \pi \) is a constant (approximately 3.14159), and \( r \) is the radius of the circle.
This equation is useful because it shows the relationship between the radius and the circumference. By manipulating this equation, as seen in the solution steps, we can solve for different variables given the value of another. This principle is foundational in algebra.
Variables
Variables are symbols used in mathematics to represent numbers in equations. They allow us to express generalized formulas that can apply to many situations.
In this exercise, \( r \) and \( C \) are variables, with \( r \) representing the radius and \( C \) the circumference.
By understanding the role of variables, you can see how changing one variable affects another. In our example, increasing the radius will result in an increase in circumference, showing the direct relationship captured by the equation \( C = 2\pi r \). This interplay helps solve for unknowns by aligning them with known quantities.
Circle circumference
The circumference of a circle is the total distance around the circle. It is similar to the perimeter but specific to circular shapes.
In the formula \( C = 2\pi r \), the circumference is denoted by \( C \). The formula indicates that the circumference is determined by multiplying the radius \( r \) by 2 and \( \pi \). This formula highlights how perimeter concepts apply uniquely to circles.
  • The formula emphasizes the role of \( \pi \), a crucial mathematical constant related to circles.
  • It also demonstrates the simplicity of the relationship between a circle's basic dimensions.
Knowing how to calculate the circumference allows one to understand more complex problems within geometry, such as finding a circle鈥檚 radius from the circumference, as shown in the exercise solution.

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