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Chapter 0: Problem 17

Find the area of a circle with diameter \(12 \mathrm{ft}\).

Short Answer

Expert verified
The area of the circle is \(36 \pi \) or approximately \(113.04 \mathrm{ft}^2\).

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 \[ A = \pi r^2 \] where \(A\) is the area, and \(r\) is the radius of the circle.
02

Find the Radius

The radius is half of the diameter. Given the diameter \(12 \mathrm{ft}\), the radius is \[ r = \frac{12}{2} = 6 \mathrm{ft} \].
03

Substitute the Radius into the Formula

Substitute \(r = 6 \mathrm{ft}\) into the area formula: \[ A = \pi (6)^2 \].
04

Calculate the Area

Simplify the expression to find the area: \[ A = \pi \times 36 \] Therefore, the area is \[ 36 \pi \]. If you need a numerical value, use the approximation \(\pi \approx 3.14\): \[ A = 36 \times 3.14 = 113.04 \mathrm{ft}^2 \].

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Key Concepts

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

Geometry
Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space. Understanding geometry is essential for solving problems involving circles and other geometric figures. When dealing with circles, you will often need to understand their measures such as radius, diameter, circumference, and area. In this context, we are focused on finding the area of a circle, which is a measure of the space contained within its boundary. The key formula used here is \(\text{A} = \pi r^2\), where \(\text{A}\) represents the area and \(r\) is the radius of the circle.
Circle Radius
The radius of a circle is the distance from the center of the circle to any point on its circumference. It is a crucial measure because many calculations involving circles use the radius. In the exercise, the diameter of the circle is given as \(12 \mathrm{ft}\). The radius is half of the diameter. So, we calculate the radius as follows:
\[ r = \frac{12}{2} = 6 \mathrm{ft} \].
This means the circle's radius is \(6 \mathrm{ft}\).
Circle Diameter
The diameter of a circle is the straight line passing through the center of the circle, connecting two points on the circumference. It is twice the length of the radius. This means if you know either the diameter or the radius, you can easily find the other by these relationships:
\[ d = 2r \] and \[ r = \frac{d}{2} \]
For a diameter of \(12 \mathrm{ft}\), we calculated the radius as \(6 \mathrm{ft}\). This knowledge allows us to substitute the radius value in the area formula to find the area of the circle.
蟺 Approximation
The symbol \(\pi\) (pi) is a mathematical constant that represents the ratio of a circle's circumference to its diameter. Its value is approximately \(3.14\), but it is an irrational number, which means it has an infinite number of decimal places without repeating. For practical purposes, we often use the approximation \(\pi \approx 3.14\).
In our step-by-step calculation to find the area of the circle with a radius of \(6 \mathrm{ft}\), we first find:
\[ A = \pi \times (6)^2 = 36 \pi \]
When substituting the approximation for \(\pi\), we get:
\[ A = 36 \times 3.14 = 113.04 \mathrm{ft}^2 \]
This makes calculation easier and provides a numerical area value.

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Most popular questions from this chapter

In Exercises \(21-26,\) round your answer to the nearest tenth where necessary. Shawna is planning a trip in which she drives from city A to city B to city C and then returns to city A. On a map that uses a scale where 1 in. represents 50 miles, she finds that city \(A\) is 1.8 in. from city \(B\), which is 2.2 in. from city \(C\), which is 0.9 in. from city A. Find the actual driving distance for this trip.

Find the perimeter and area of the given figure. A square with diagonal 15 meters.

A side of one polygon is \(15 \mathrm{ft}\) and the corresponding side of a second similar polygon is 2 ft. If the perimeter of the first polygon is 80 ft and its area is 2000 sq fi. find the perimeter and area of the similar polygon.

Find the perimeter and area of a square with side \(8 \mathrm{ft}\).

Find the length of the side of a square with diagonal \(5 \mathrm{cm} .\)
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