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Chapter 0: Problem 18

Find the area of a circle with diameter 9 meters.

Short Answer

Expert verified
The area is approximately 63.62 square meters.

Step by step solution

01

Identify the diameter

The diameter of the circle is given as 9 meters.
02

Calculate the radius

The radius is half of the diameter. So, the radius is \( \frac{9}{2} = 4.5 \) meters.
03

Use the area formula

The formula for the area of a circle is \[ A = \pi \times r^2 \]. Substituting the radius \ (r = 4.5) \ into the formula, we get \[ A = \pi \times (4.5)^2 \]
04

Calculate the area

Simplify the expression to find the area: \[ A = \pi \times 20.25 \approx 63.62 \] meters squared.

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Key Concepts

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

geometry
Geometry is the branch of mathematics that deals with shapes, sizes, and properties of space. Applications of geometry are found in various fields such as engineering, architecture, and art. In this exercise, we focus on the properties of circles, a fundamental shape in geometry. A circle is defined by its set of points equidistant from a center point. Key measurements in circles include the diameter and the radius.
The diameter is the longest distance across the circle, passing through the center, and the radius is half of the diameter.
Understanding these properties allows us to apply formulas to find important measurements, like the area or circumference of a circle.
circle radius
The radius of a circle is the distance from the center to any point on the edge. It is a crucial measurement because many formulas involving circles use the radius as a key component.
In our exercise, the diameter of the circle is given as 9 meters. To find the radius, we simply divide the diameter by 2. This yields: \( \text{radius} = \frac{9}{2} = 4.5 \) meters.
Knowing the radius allows us to further calculate other properties of the circle, such as the area and circumference.
area formula
The formula to calculate the area of a circle is essential in geometry. It allows us to determine the amount of space inside the circle. The formula is expressed as:
\( A = \pi \times r^2 \)
Where \( A \) represents the area, \( \pi \) (pi) is a constant approximately equal to 3.14159, and \( r \) is the radius of the circle.
In our specific exercise:
  • The radius \( r = 4.5 \) meters
  • Substitute the radius into the formula: \( A = \pi \times (4.5)^2 \)
  • Calculate the area: \( A = \pi \times 20.25 \)
  • Approximate the area: \( A \approx 63.62 \) square meters

This comprehensive understanding of the area formula helps us solve problems involving the space a circle occupies. Understanding the steps ensures students can apply the formula confidently in various situations.

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Most popular questions from this chapter

Find the circumference of a circle with radius \(4 \mathrm{ft}\).

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

Find the arc length of a \(30^{\circ}\) sector of a circle with a radius of 9 in.

Answers should be rounded to the nearest tenth unless otherwise indicated. A closed right circular cylinder has a radius of \(3 \mathrm{m}\). Find the volume of the cylinder if its lateral surface area is \(96 \pi\) square meters. Leave your answer in terms of \(\pi\).

Find the perimeter and area of the given figure. A rectangle with height 5 in. and diagonal 9 in.
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