Problem 120 - Algebra and Trigonometry Real Mathematics, Real People

Question

Prove that the area of a circular sector of radius \(r\) with central angle \(\theta\) is \(A=\frac{1}{2} r^{2} \theta,\) where \(\theta\) is measured in radians.

Step-by-Step Solution

Answer

The area of a circular sector with radius \(r\) and central angle \(\theta\) (in radians) is \(\frac{1}{2} r^{2} \theta\).

Step 1: Understanding the Problem

A circular sector is the region of a circle enclosed by two radii and their intercepted arc. Here, we are considering the radius of the sector as \(r\), and the central angle as \(\theta\). We are required to prove that the area of this sector \(A=\frac{1}{2} r^{2} \theta\), where \(\theta\) is in radians.

Step 2: Area of the Whole Circle

The area \(A\) of a circle with radius \(r\) is given by the formula \(A=\pi r^{2}\). This is the entire area enclosed by the circle.

Step 3: Area of the Circular Sector

The fraction of the circle that the sector represents is \(\frac{\theta}{2\pi}\) because the whole circle represents \(2\pi\) radians. So the area \(A_{\text{sector}}\) of the sector is given by the formula \(A_{\text{sector}} = \frac{\theta}{2\pi} (\pi r^{2})\). Simplify this expression to obtain \(A_{\text{sector}} = \frac{1}{2} r^{2} \theta\).

Key Concepts

Circular Sector

In geometry, a circular sector is a part of a circle. It looks like a slice of pizza or a pie. Imagine a full circle that you cut into smaller parts. Each part made by two straight lines (radii) and the connected curve (arc) is called a sector. The size of the sector depends on the angle between the two radii, called the central angle, and the radius of the circle.

• A circular sector is bounded by two radii and an arc.
• The sector's area can vary depending on the central angle's size.

Understanding circular sectors helps when dealing with pie charts, clocks, and any circular objects that have parts or segments.

Central Angle

The central angle is a key part of defining a circular sector. This angle is formed at the center of the circle by the two radii. It determines how large the pie-like slice will be.

• It plays a crucial role in calculating the sector's area.
• The central angle also determines the length of the arc that makes up part of the sector's boundary.

The measurement of the central angle is essential for determining the sector's properties, such as its area. It is this angle that helps us understand how much of the total circle the sector represents.

Radians

Radians are a way of measuring angles, just like degrees. However, radians are often used in higher mathematics because they simplify many formulas. Unlike degrees, which divide a circle into 360 parts, radians divide it into approximately 6.28 parts (or 2π).

• One full circle is equivalent to 2π radians.
• This makes calculations involving angles and circles more straightforward in many cases.

When dealing with a circular sector, it's typical to express the central angle in radians. This is because it directly relates the angle to the radius and arc length, simplifying calculations such as finding the area of the sector using the formula: \[A = \frac{1}{2} r^2 \theta\] where \( \theta \) is the angle in radians.

Circle Area

The area of a circle is a fundamental concept in geometry. It is given by the formula \(A = \pi r^2\), where \(\pi\) (pi) is a mathematical constant approximately equal to 3.14159, and \(r\) is the circle's radius.

• The area represents the amount of space enclosed by the circle.
• Knowing the circle's area is crucial when working with circular sectors.

In the context of circular sectors, understanding the full area of the circle helps us calculate what fraction of the circle the sector occupies. This is how we derive the formula for the area of the sector through the expression: \[A_{\text{sector}} = \frac{\theta}{2\pi} (\pi r^2)\]which simplifies to:\[A_{\text{sector}} = \frac{1}{2} r^2 \theta\] This formula shows how the area is proportionate to the central angle, \(\theta\), when expressed in radians, alongside the radius squared.