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Chapter 5: Problem 9

Use 3.14 for \(\pi\). Approximate the area of a circle whose diameter is 20 ft.

Short Answer

Expert verified
The area of the circle is approximately 314 square feet.

Step by step solution

01

Understand the formula

To calculate the area of a circle, use the formula: \[ A = \pi r^2 \]
02

Determine the radius

The radius (\r) is half the diameter. Since the diameter is 20 ft, divide by 2:\[ r = \frac{20}{2} = 10 \text{ft} \]
03

Substitute the radius and \pi

Using \pi \approx 3.14 and r = 10 ft, substitute the values into the formula:\[ A = 3.14 \cdot (10)^2 \]
04

Calculate the area

Square the radius and then multiply by \pi:\[ A = 3.14 \cdot 100 = 314 \text{square feet} \]

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Key Concepts

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

area of a circle
The area of a circle represents the amount of space enclosed within its boundary. It's a specific region that we calculate using a mathematical formula. The formula to compute the area is given by: \( A = \pi r^2 \). Here, \( A \) stands for the area, pi (\( \pi \)) is a constant approximately equal to 3.14, and \( r \) is the radius of the circle.
Understanding the area is crucial in various real-life contexts, like determining the space you can cover with a circular rug or the surface area of a pizza. By knowing how to calculate the area, you can make informed decisions and solve practical problems.
radius and diameter
The radius and diameter are essential properties of a circle. The radius is the distance from the center of the circle to its boundary. The diameter is the distance across the circle, passing through the center, and is twice the radius. Mathematically, we express this relationship as: \( d = 2r \).
In this exercise, we were provided with the diameter of the circle, which was 20 feet. To find the radius, we simply divided the diameter by 2: \( r = \frac{20}{2} = 10 \) feet.
Knowing how to derive one from the other is vital for solving problems related to circles, as many formulas, including the area formula, require the radius.
use of pi (蟺)
Pi (\( \pi \)) is a unique and fascinating mathematical constant essential for calculations involving circles. It represents the ratio of a circle's circumference to its diameter, which remains constant for all circles. The value of \( \pi \) is approximately 3.14, although it is an irrational number with an infinite number of non-repeating decimal places.
In area calculations, \( \pi \) is used to ensure the formula accurately represents the space within the circle. For this exercise, we approximated \( \pi \) to 3.14 to make our calculations straightforward.
Using \( \pi \) correctly allows us to compute the area accurately, ensuring our results are precise and reliable.

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