Question

Use dimensional analysis to determine which of the following expressions gives the area of a circle: Is it \(\pi r^{2}\) or \(2 \pi r\) ? Explain.

Short answer

\(\pi r^2\) is the correct expression for the area of a circle.

Step-by-step solution

Understand the Problem

We need to determine which expression, either \(\pi r^2\) or \(2 \pi r\), correctly represents the area of a circle using dimensional analysis. Dimensional analysis involves checking if the dimensions of an expression match the expected dimensions of the quantity it represents.

Identify the Dimensions

Identify the dimensions of the quantities involved. The area of a circle should have dimensions \([L^2]\), where \(L\) is a length dimension. The radius \(r\) has the dimension of length \([L]\).

Analyze \(\pi r^2\)

The term \(\pi\) is a dimensionless constant, so it does not affect the dimensions. The expression \(r^2\) has dimensions \([L^2]\), because \(r\) is length \([L]\) and multiplying \(r\) by itself gives \([L^2]\). Thus, \(\pi r^2\) has dimensions \([L^2]\), which matches the dimensions of area.

Analyze \(2\pi r\)

The term \(2\pi\) is dimensionless. The term \(r\) has dimensions \([L]\), so the whole expression \(2\pi r\) has dimensions \([L]\). Since this is not \([L^2]\), it does not match the dimensions required for an area.

Determine the Correct Expression

Based on dimensional analysis, \(\pi r^2\) has the correct dimensions \([L^2]\) for an area. Therefore, \(\pi r^2\) correctly represents the area of a circle, while \(2\pi r\) does not.

Key concepts

Area of a Circle

The area of a circle is a fundamental concept in geometry that helps us understand the space within a circular boundary. To calculate the area, we use the formula \[ A = \pi r^2 \]where \(A\) represents the area, \(\pi\) is a constant approximately equal to 3.14159, and \(r\) is the radius of the circle. The radius is the distance from the center of the circle to any point on its circumference.

If you imagine a circle, the area can be thought of as the number of unit squares that fit inside the circle. This formula captures that idea succinctly by using the radius as a scaling factor. Knowing the area allows us to describe the size of the circle in terms of square units, which is crucial in many applications such as engineering, architecture, and everyday measurements.

Dimensional Consistency

Dimensional consistency is a key principle in physics and mathematics, ensuring that equations and expressions make sense in terms of their units.

In the context of the circle's area, we're interested in ensuring that our expressions relate to areas, which have dimensions of length squared \([L^2]\).

• When you see \(\pi r^2\), the \(\pi\) is dimensionless, and \(r\) (radius) has dimensions of length \([L]\).
• Multiplying \(r\) by itself gives us \(r^2\), which has dimensions \([L^2]\).

Thus, \(\pi r^2\) is dimensionally consistent with an area. On the other hand, the expression \(2\pi r\) has dimensions of just length \([L]\), reflecting a linear measurement like a circle's circumference.

Ensuring dimensional consistency helps verify that our equations are constructed correctly and align with the physical or mathematical concepts they represent.

Mathematical Expressions

Mathematical expressions like \(\pi r^2\) and \(2\pi r\) are shorthand notations used to convey complex ideas or calculations effortlessly. A strong grasp on interpreting these expressions is essential for solving problems efficiently.

• \(\pi r^2\): This expression is compact yet powerful, summarizing the calculation needed to determine a circle's area.
• \(2\pi r\): While dimensionally inconsistent for area, this expression correctly represents a circle’s circumference — the perimeter or boundary length.

Interpreting these expressions involves understanding the role each symbol plays, especially constants like \(\pi\) and variables like \(r\). By understanding each element's purpose and how they combine to form meaningful calculations, we can apply these expressions confidently in various scenarios.

This skill is not only valuable in coursework but is also crucial in many real-life applications where mathematical reasoning supports decision-making and problem-solving.