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Chapter 8: Problem 1

Find the exact circumference and area of a circle whose radius has length 8 cm.

Short Answer

Expert verified
The circumference is \(16\pi\) cm and the area is \(64\pi\) cm虏.

Step by step solution

01

Understand the Formula for Circumference

The circumference of a circle is given by the formula \( C = 2\pi r \), where \( C \) is the circumference and \( r \) is the radius.
02

Calculate the Circumference

Substitute the radius \( r = 8 \) cm into the circumference formula: \( C = 2\pi \times 8 \). Simplify to get \( C = 16\pi \) cm.
03

Understand the Formula for Area

The area of a circle is given by the formula \( A = \pi r^2 \), where \( A \) is the area and \( r \) is the radius.
04

Calculate the Area

Substitute the radius \( r = 8 \) cm into the area formula: \( A = \pi \times 8^2 \). First, compute \( 8^2 = 64 \), and then, \( A = 64\pi \) cm虏.

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Key Concepts

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

Circumference of a Circle
When we talk about the circumference of a circle, we are referring to the distance around the circle. Think of it like the perimeter, but for circles. To compute the circumference, we use the formula: \( C = 2\pi r \). This formula tells us that the circumference \( C \) is equal to twice the product of \( \pi \) (approximately 3.14159) and the radius \( r \) of the circle.

This formula is key because it uses the constant \( \pi \), which relates the circumference of any circle to its diameter. Since the diameter is twice the radius, the multiplication by 2 in the formula makes it all come together perfectly.
  • Example: if the radius \( r \) is 8 cm, we substitute it into the formula: \( C = 2\pi \times 8 \).
  • After calculation, the circumference is \( 16\pi \) cm, which gives you the distance around the entire circle.
Area of a Circle
The area of a circle gives us the amount of space inside the circle. Visualize it as the surface that you would "paint" inside the boundary of the circle. The formula used for the area is \( A = \pi r^2 \).

This formula involves squaring the radius \( r \) of the circle and then multiplying by \( \pi \) to calculate the area \( A \). Squaring the radius accounts for both the width and the height of the circle's space, hence why \( r^2 \) gives you the 'square' units of area covered.
  • For instance, with a radius of 8 cm, substitute into the formula: \( A = \pi \times (8)^2 \).
  • Calculating, \( 8^2 = 64 \), which results in an area \( A = 64\pi \) cm虏, representing the total region you cover within the edge of the circle.
Radius
The radius of a circle is a fundamental concept in circle geometry. It represents the distance from the center of the circle to any point on its edge. It's a straight line that connects the center of the circle to the perimeter.

The radius is crucial because it helps in defining the size of the circle and is used in all primary calculations related to circles鈥攊ncluding circumference and area. The diameter, which is twice the radius, makes understanding the radius even more important, as many circle-related formulas leverage these two dimensions.
  • If given or measured, the radius allows for straightforward computation of other circle properties. In our example, with a radius of 8 cm, many other properties, like diameter (16 cm) or the circumference, can be directly calculated.
  • Grasping the radius helps set a foundation for exploring more complex geometrical concepts.

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Most popular questions from this chapter

In a triangle with sides of lengths \(a, b,\) and \(c\) and semiperimeter \(s\), show that the length of the radius of the inscribed circle is $$r=\frac{2 \sqrt{s(s-a)(s-b)(s-c)}}{a+b+c}$$

Find the exact lengths of the radius and the diameter of a circle whose area is: a) \(25 \pi\) in \(^{2}\) b) \(2.25 \pi \mathrm{cm}^{2}\)

A rectangle has a perimeter of 16 in. What is the limit (largest possible value) of the area of the rectangle?

Find the approximate perimeter of a regular polygon that has 20 sides if the length of its radius is \(7 \mathrm{cm}\)

Use your calculator value of \(\pi\) unless otherwise stated. Round answers to two decimal places. At center court on a gymnasium floor, a large circular emblem is to be painted. The circular design has a radius length of 8 ft. a) What is the area to be painted? b) If a quart of paint covers \(70 \mathrm{ft}^{2},\) how many quarts of paint are needed to complete the job? c) If each quart of paint costs \(\$ 15.89,\) find the cost of the paint needed.
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