Question

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

Short answer

Question: Prove that the area of a circular sector with radius r and central angle θ (measured in radians) is A = (1/2) * r^2 * θ. Answer: Following the step by step solution provided, we first established the area of a full circle with radius r as $\pi r^2$. Then, we found the fraction of the circle covered by the sector by dividing the central angle in radians (θ) by the total angle for a full circle (2π) and calculated the area of the sector by multiplying the area of the full circle by this fraction. Finally, we simplified the formula to derive the area of a sector: $A=\frac{1}{2}{r}^2\theta$.

Step-by-step solution

Understand the area of a full circle

The area of a full circle with radius r can be calculated using the formula $A=\pi r^2$. In our exercise, we want to find the area of the sector with central angle θ, which is just a part of the full circle.

Convert the central angle to radians

In order to find the area of the sector, the central angle must be measured in radians. If the central angle θ is given in degrees, convert it to radians by multiplying it by $\frac{\pi }{180}$, as follows: ${\theta }_{radians}={\theta }_{degrees}\times \frac{\pi }{180}$ In this exercise, we are given that the central angle is already in radians, so we can skip this step.

Find the fraction of the circle covered by the sector

To find the fraction of the circle that the sector represents, we need to divide the central angle θ (in radians) by the total angle for a full circle, which is $2\pi$ radians: $f=\frac{\theta }{2\pi }$

Calculate the area of the sector

Now that we know the fraction of the circle covered by the sector, we can determine the area of the sector by multiplying the area of the full circle by this fraction: $A_{sector}=A_{circle}\times f=\pi r^2\times \frac{\theta }{2\pi }$

Simplify the formula

Simplify the formula for the area of the circular sector by canceling out the $\pi$ terms and multiplying the remaining terms: $A_{sector}=\pi r^2\times \frac{\theta }{2\pi }=\frac{1}{2}{r}^2\theta$ Therefore, the area of a sector with radius r and central angle θ (measured in radians) is given by $A=\frac{1}{2}{r}^2\theta$.

Key concepts

Area of a sector

In geometry, the area of a sector is a fascinating way to understand a fraction of a circle's area. Imagine slicing a pizza or pie and the wedge that you remove is what we call a sector. To calculate the area of this specific part, we need the circle's radius, which is the distance from the center to any point on the circle, and the central angle of the sector, which is the angle that the sector makes at the circle's center.
The area of a full circle with radius $r$ is calculated using the well-known formula $A=\pi r^2$. A sector is a fraction of the full circle, and its area is proportional to the central angle over the whole circle's angle. The formula for the area of a sector becomes $A=\frac{1}{2}{r}^2\theta$. Here, $\theta$ is the central angle measured in radians, which we will explore further.
This sector area formula is derived by multiplying the total area of the circle by $\frac{\theta }{2\pi }$, the fraction of the circle made by the sector. The derivation is straightforward once you grasp that a sector's area is part of the total circle's area, based on the ratio of its angle to the entire circle's angle.

Radians

Understanding radians is crucial in calculating any parameter related to circles, such as the area of a sector. When we think of radians, we are talking about a way of measuring angles based on the radius of the circle.
Radians offer a natural angle measurement, where one radian is the angle made when the arc length of a circle equals the circle's radius. A full circle contains $2\pi$ radians, which equates to 360 degrees. Therefore, one important conversion to remember is that one radian is approximately 57.3 degrees. These conversions are often needed when solving circle-related problems in calculus and geometry.
In most mathematical contexts, especially in calculus, radians are preferred because they simplify formulas significantly. For example, the derivative of sine and cosine functions rely on angle measures in radians. Plus, they make the computation of trigonometric and calculus-related problems much more straightforward.

Circle geometry

Circle geometry forms a foundation for understanding various geometric principles and theorems. When studying circles, we focus on properties like the radius, diameter (which is twice the radius), circumference $C=2\pi r$, and area $A=\pi r^2$.
A central angle in circle geometry refers to an angle whose vertex is the center of the circle, and its sides are formed by two radii. This angle allows us to calculate both the arc length and sector area directly. To find an arc's length for instance, multiply the radius by the central angle in radians. Similarly, sector area calculations use the same concept where the center's subtended angle determines a fraction of the circle area.
Circle geometry extends into various applications across math and science. Understanding these concepts aids in more complex applications such as calculating the length of elliptic curves, planetary motion in orbits, and many engineering tasks where curves design is essential.