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Chapter 20: Problem 379

Find the area of a circle whose radius is 7 in. [Use \(\pi=22 / 7]\).

Short Answer

Expert verified
The area of the circle with a radius of 7 inches is 154 square inches.

Step by step solution

01

Identify the given values and the formula

We are given the radius (r) as 7 in, and 蟺 as 22/7. The formula for the area of a circle is: Area = 蟺 * r^2
02

Substitute the given values into the formula

Now, plug the given values of radius and 蟺 into the formula: Area = (22/7) * (7)^2
03

Perform the calculations

Calculate the area by squaring the radius and multiplying by the given value of 蟺: Area = (22/7) * (49) Area = 22 * 7
04

Simplify the result

Finally, multiply 22 by 7 to find the area: Area = 154 sq. in So, the area of the circle with a radius of 7 inches is 154 square inches.

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Key Concepts

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

Understanding the Radius
The radius of a circle is a crucial component when calculating its area. The radius is the distance from the center of the circle to any point on its edge. In simple terms, if you imagine a straight line drawn from the center to the circle's boundary, that line represents the radius.

In our exercise, the radius is given as 7 inches. This means every point on the circle's edge is exactly 7 inches away from the center. Understanding the radius is essential because it is used in geometric formulas, especially when calculating the area of a circle.
  • **Radius in Formulas**: It is commonly represented by the letter \( r \).
  • **Importance**: Knowing the radius helps in finding the size of the circle and other properties, such as the circumference.
Exploring Pi Value
Pi, often denoted by the symbol \( \pi \), is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It's an irrational number, meaning it has an infinite number of decimal places without repeating.

In our exercise, the value of \( \pi \) is approximated to \( \frac{22}{7} \). This approximation helps simplify calculations, especially when dealing with whole numbers. However, \( \pi \) is approximately 3.14159 in decimal form.
  • **Why Use 22/7?**: It offers a simple fraction that closely approximates \( \pi \).
  • **Role in Geometry**: \( \pi \) is crucial in formulas relating to circles, such as the area and circumference.
Understanding \( \pi \) aids in calculating not just the area, but also in other geometrical aspects of circles.
Understanding Geometric Formulas
Geometric formulas are the foundation of finding measurements and properties of shapes. For circles, these formulas describe various attributes like area and circumference. In our scenario, we are focusing on the formula to find the area of a circle:
\[ \text{Area} = \pi \times r^2 \]
This formula tells us that the area (the amount of space inside the circle) is calculated by multiplying \( \pi \) by the square of the radius. Essentially, we are finding how many square units fit inside the circle.
  • **Why Square the Radius?**: Squaring the radius scales the one-dimensional radius into a two-dimensional area measurement.
  • **Importance in Geometry**: These formulas help in converting abstract concepts into tangible measurements.
Mastering these formulas allows you to easily find the area and other geometric properties of circles and other shapes.

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Most popular questions from this chapter

If the areas of the two circles are 48 and 75, find the ratio of their circumferences.

The ratio of the areas of two circles is \(9: 4\). The length of the radius of the larger circle is how many times greater than the length of the radius of the smaller circle?

The ratio of the area of two circles is \(16: 1\). If the diameter of the smaller circle is 3 , find the diameter of the larger circle.

A farmer is cutting a field of oats with a machine which takes a \(5 \mathrm{ft}\). cut. The field he is cutting is circular and when he has been round it \(11(1 / 2)\) times (starting from the perimeter) he calculates that he has cut half the area of the field. How large is the field? (Answer to the nearest 100 sq. yds.)

The area of circle 0 is 36 sq. in. If there are \(40^{\circ}\) contained in central angle \(\mathrm{COD}\), find the number of square inches In the area of sector COD.
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