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Chapter 2: Problem 10

Express the area, \(A\), of a circle as a function of its circumference, \(C\).

Short Answer

Expert verified
The area is \( A = \frac{C^2}{4\pi} \).

Step by step solution

01

Understanding the Relationship

A circle's circumference is given by the formula \( C = 2\pi r \), where \( r \) is the radius. The area is given by \( A = \pi r^2 \). Our goal is to express \( A \) in terms of \( C \).
02

Solve for Radius

From the circumference formula, \( C = 2\pi r \), solve for the radius \( r \). This gives us \( r = \frac{C}{2\pi} \).
03

Substitute Radius into Area Formula

Substitute \( r = \frac{C}{2\pi} \) into the area formula \( A = \pi r^2 \). This becomes \( A = \pi \left( \frac{C}{2\pi} \right)^2 \).
04

Simplify the Area Expression

Simplify \( A = \pi \left( \frac{C}{2\pi} \right)^2 \) by calculating \( \left( \frac{C}{2\pi} \right)^2 \): \[ \left( \frac{C}{2\pi} \right)^2 = \frac{C^2}{4\pi^2} \]Substitute back into the formula: \[ A = \pi \cdot \frac{C^2}{4\pi^2} \]Further simplify to get:\[ A = \frac{C^2}{4\pi} \]
05

Final Expression

The area \( A \) of a circle as a function of its circumference \( C \) is \( A = \frac{C^2}{4\pi} \).

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Key Concepts

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

Circumference
The circumference of a circle is like the perimeter we find in polygons. It measures the distance around the circle. To calculate the circumference ( C ), you use the formula: C = 2\pi r . Here, \( r \) represents the circle's radius, and \( \pi \) (pi) is a special mathematical constant approximately equal to 3.14159.
  • Role of Pi: Pi helps us understand the relationship between a circle's diameter and its circumference.
  • Why Twice the Radius?: Since the radius is half the diameter, multiplying the diameter by \( \pi \) accounts for half the size twice.
  • Applications: This formula is useful in finding how long the boundary of a circle is, such as the length of a circular track.
The simplicity of the formula belies the importance of the concept; it's a foundational element of circle geometry.
Area of a Circle
The area of a circle tells us how much space is enclosed within its boundary. We calculate it using the formula: A = \pi r^2 , where \( r \) is the circle's radius.
  • Pi Again: Just like with the circumference, \( \pi \) is crucial here and relates the square of the radius to the area.
  • The Exponent: Squaring the radius ( r^2 ) shows how area grows with the radius; a little change in the radius makes a big change in the area.
  • Visualization: Imagine trying to lay small tiles to fill the circle. This tile count is the area.
You come across the area formula in various practical scenarios, from calculating the space on circular tables to understanding crop circles in agriculture.
Radius
The radius of a circle is the distance from its center to any point along its edge. It's a crucial measure in circle geometry. If you think of a circle as a wheel, the radius is like the spoke reaching from the center to the rim.
  • Connecting Concepts: The radius links the circle's circumference and area. For instance, both principal formulas for circumference and area use the radius.
  • Expressing the Radius: From the circumference formula, \( C = 2\pi r \), you can determine the radius as \( r = \frac{C}{2\pi} \).
  • Everyday Geometry: Knowing the radius helps solve practical problems, like finding the right size of a circular carpet or a pizza of a specific size.
The radius is more than a measure; it's the key to unraveling many geometric secrets of circles.

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Most popular questions from this chapter

Find the inverse function \(f^{-1}\) and state its domain. $$f(x)=x^{3}+1$$

For each pair of functions,\( find \)(f \circ g)(x)\( and \)(g \circ f)(x)$ and state the domain for each. $$f(x)=\frac{x}{x-1}, g(x)=x^{2}-1$$

Specify a sequence of transformations to perform on the graph of \(y=x^{2}\) to obtain the graph of the given function. $$p: x \mapsto \frac{1}{2}(x+4)^{2}$$

Find the domain of the function. $$\\{(-6.2,-7),(-1.5,-2),(0.7,0),(3.2,3),(3.8,3)\\}$$

Specify a sequence of transformations to perform on the graph of \(y=x^{2}\) to obtain the graph of the given function. $$g: x \mapsto(x-3)^{2}+5$$
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