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Chapter 2: Problem 5

If the diameter of a circle is \(10 \mathrm{ft}\), what is the radius?

Short Answer

Expert verified
The radius of the circle is 5 ft.

Step by step solution

01

Understand the relationship between diameter and radius

The diameter of a circle is twice the length of its radius. Mathematically, this relationship can be expressed as: \[ d = 2r \] where \( d \) is the diameter and \( r \) is the radius.
02

Identify the given value

In the problem, the given value is the diameter of the circle, which is \(10 \text{ ft} \).
03

Rearrange the formula to solve for the radius

To find the radius, rearrange the formula \( d = 2r \) to solve for \( r \). Divide both sides by 2: \[ r = \frac{d}{2} \]
04

Substitute the given diameter into the formula

Substitute the given diameter value of \( 10 \text{ ft} \) into the formula: \[ r = \frac{10}{2} \]
05

Compute the radius

Calculate the value: \[ r = 5 \text{ ft} \]

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Key Concepts

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

diameter and radius relationship
The relationship between the diameter and radius of a circle is fundamental in circle geometry. If you know one, you can always find the other. The diameter is simply twice the radius. This straightforward relationship is expressed by the formula:

\( d = 2r \)
where \( d \) stands for diameter and \( r \) stands for radius.

Conversely, if you have the diameter and need to find the radius, you just need to divide the diameter by 2. Mathematically, you rearrange the original formula to solve for \( r \):
\( r = \frac{d}{2} \)
This simple relationship helps in many geometric calculations and is essential for understanding circle properties.
basic algebra
Basic algebra is crucial when working with formulas in geometry. When faced with an equation like \( d = 2r \), you can rearrange it to solve for any variable. If given the diameter, you can find the radius by isolating \( r \).

Here鈥檚 how you do it step-by-step:
  • Start with the original equation:\
circle geometry
Understanding circle geometry involves knowing key properties and relationships within a circle. Two of the most important measurements are the diameter and the radius.

Some core properties of these measurements include:
  • The diameter is the longest straight line that passes through the center of the circle, touching the circumference at two points.

  • The radius is half of the diameter and extends from the center of the circle to any point on the circumference.

These properties help in solving various geometric problems, such as finding the area \( A \) or circumference \( C \) of a circle:
  • The area is given by \( A = \pi r^2 \)

  • The circumference is \( C = 2 \pi r \).

So, understanding the diameter and radius relationship is not just about solving simple equations; it is about forming the foundation for broader geometric calculations.鈥�

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