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Chapter 2: Problem 18

Solve each formula for the indicated letter. \(A=\pi r^{2},\) for \(r^{2}\)

Short Answer

Expert verified
The formula solved for \(r^{2}\) is \frac{A}{\type1{\pi}}.

Step by step solution

01

Understand the Given Formula

The given formula is the area of a circle: ewline A = \pi r^{2}ewline The task is to solve for the term \(r^{2}\).
02

Isolate the Term

To isolate \(r^{2}\), divide both sides of the equation by \pi:ewline \frac{A}{\type1{\pi}} = r^{2}
03

Simplify the Right Side

The right side \(r^{2}\) is already simplified, so the formula becomes:ewline \boxed{r^{2} = \frac{A}{\type1{\pi}}}

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Key Concepts

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

area of a circle
When we talk about the area of a circle, we refer to the space contained within the boundaries of the circle. The formula to find the area of a circle is given by: \(A = \pi r^{2}\).
Here,
  • \(A\) is the area of the circle
  • \(\pi\) is a constant approximately equal to 3.14159
  • \(r\) is the radius of the circle, which is the distance from the center of the circle to any point on its circumference

The symbol \(\pi\) represents the ratio of the circumference of a circle to its diameter. This formula is essential in finding how much space is occupied by the circle.
algebraic manipulation
Algebraic manipulation is the technique of rearranging equations to isolate a specific variable. It involves using operations like addition, subtraction, multiplication, and division to change the form of an equation. Let's break down the manipulation applied to our area of a circle formula:
To isolate \(r^{2}\) from the formula \(A=\pi r^{2}\), follow these steps:
  • First, identify the term that encloses \(r^{2}\). Here, it鈥檚 \(\pi\).
  • Next, perform the opposite operation that is applied to \(r^{2}\). Since \(\pi\) is multiplied by \(r^{2}\), you will divide both sides of the equation by \(\pi\). This step is key in isolating the variable of interest.
By dividing both sides by \(\pi\):
\(\frac{A}{\pi} = r^{2}\)
This equation tells us that the area divided by \(\pi\) equals the square of the radius. Such manipulation simplifies complex equations and makes them easier to solve.
isolating variables
Isolating variables is a fundamental skill in algebra where you manipulate an equation to get one specific variable by itself on one side of the equation. This is particularly useful when you need to solve for that variable. Here's how it works in our example:
Given the formula for the area of a circle: \(A = \pi r^{2}\)
To isolate \(r^{2}\), you need to rearrange the formula. Follow these steps:
  • Start with the equation: \(A = \pi r^{2}\)
  • To get \(r^{2}\) on its own, divide both sides by \(\pi\): \(\frac{A}{\pi} = r^{2}\)
  • Now, you have successfully isolated \(r^{2}\).

Isolating variables allows you to simplify an equation and find the value of a specific term. This process is crucial in solving many types of mathematical problems. By understanding how to isolate variables, you can tackle more complex equations with confidence.

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