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Chapter 4: Problem 1

Translate the following sentence into a mathematical equation: The area \(A\) of a circle equals the product of the square of its radius \(r\) and the constant \(\pi\).

Short Answer

Expert verified
A = \pi r^2

Step by step solution

01

Identify the given variables

The problem mentions the area of a circle, denoted as \(A\), and the radius of the circle, denoted as \(r\).
02

Recognize the constant

The problem also mentions the constant \(\backslashpi \) which is approximately equal to 3.14159.
03

Understand the relationship

The problem states that the area \(A\) is equal to the product of the square of the radius \(r^2\) and the constant \(\backslashpi \).
04

Translate into a mathematical equation

Combining all the information, the mathematical equation becomes: \[ A = \pi \cdot r^2 \]

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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 measures the amount of space inside the circle's boundary. This concept is essential for geometry and various fields that require space calculation. To determine the area, we use the formula involving the radius of the circle and a mathematical constant, \( \pi \) (pi). Calculating the area is important for understanding sizes and quantities in real-life applications, from designing circular objects to calculating land areas. A basic formula represents this area calculation, and understanding it helps grasp fundamental geometric principles.
radius
The radius of a circle is the distance from its center to any point on its circumference. It is a crucial component for calculating other properties of the circle, including the area and circumference. The radius connects directly to the formula for the area of a circle: the area depends on the square of the radius, making it a vital measure for geometric calculations. Understanding the radius concept helps in comprehending how sizes change with various radius lengths and their impact on the area and other attributes.
pi
Pi (\( \pi \)) is an essential constant in mathematics related to circles. It approximates to 3.14159 and represents the ratio of the circumference of any circle to its diameter. This constant appears in various mathematical formulas, including the area of a circle. Incorporating pi into equations allows for precise calculations of circle-related measurements. The value of pi remains consistent and serves as a universal constant that simplifies complex geometric computations.
mathematical equation translation
Translating mathematical sentences into equations involves identifying variables, constants, and their relationships. This transformation simplifies complex word problems into understandable mathematical expressions. For example, to find the area of a circle, we identify the area (\(A\)), the radius (\(r\)), and the constant pi (\( \pi \)). The sentence 'The area of a circle equals the product of the square of its radius and the constant pi鈥� translates into the equation \( A = \pi \cdot r^2 \). Understanding how to translate these sentences is fundamental to solving mathematical problems efficiently.

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Most popular questions from this chapter

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