Problem 33 - Technical Mathematics with Calculus

Question

Earlier we gave the formula for the area of a circular sector of radius \(r\) and central angle \(\theta:\) area \(=r^{2} \theta / 2\) (Eq. 77 ). Using Eq. \(76, \theta=s / r,\) show that the area of a sector is also equal to \(r s / 2,\) where \(s\) is the length of the arc intercepted by the central angle.

Step-by-Step Solution

Answer

The area of a sector can also be expressed as \(\frac{rs}{2}\) by substituting \(\theta = \frac{s}{r}\) into the original area formula and simplifying.

Step 1: Identify given formulas

Recognize that the area of a circular sector is given by the formula: Area = \(\frac{r^{2} \theta}{2}\) (Eq. 77) and relate the central angle \(\theta\) to the arc length \(s\) and radius \(r\) using: \(\theta = \frac{s}{r}\) (Eq. 76).

Step 2: Substitute the central angle expression

Substitute the expression for \(\theta\) from Eq. 76 into the area formula from Eq. 77. Replace \(\theta\) with \(\frac{s}{r}\) in the area formula: Area = \(\frac{r^{2}}{2} \cdot \frac{s}{r}\).

Step 3: Simplify the expression

Simplify the equation by canceling one \(r\) in the numerator and the denominator: Area = \(\frac{r^{2}}{2} \cdot \frac{s}{r} = \frac{rs}{2}\). This results in the area of the sector formula in terms of \(r\) and \(s\).

Key Concepts

Understanding the Central Angle

The central angle is a key concept when studying the properties of a circular sector. It is the angle that is formed at the center of the circle by two radii that extend to the edges of the sector. Imagine a slice of pizza; the tip represents the center of your pizza (circle), and the crust edge of that slice corresponds to the circle's circumference. The pointed end where you would start to take a bite represents the vertex of the central angle. This angle is crucial because it helps determine the proportion of the circle that the sector represents.

Let's visualize this clearly. If a circle has a central angle of 360 degrees, the sector would actually be the entire circle. However, if the central angle were 90 degrees, then the sector would be a quarter of the circle. An important relation to remember is that the central angle, in radians, is directly proportional to the arc length it intercepts, described by the equation \(\theta = \frac{s}{r}\), where \(s\) is the arc length and \(r\) is the radius.

Arc Length and Its Implications

Arc length is the distance along the curved line that makes up the arc, which is a portion of the circumference of the circle. To make it more relatable, consider it as the measurement of the rim on your bike's wheel that the central angle 'opens up' to. If you were to take a string and lay it along the arc from one point of the sector to another, the length of that string would be the arc length. This length is larger for bigger central angles and for larger circles.

To find the arc length, you can use the formula \(s = r\theta\), where \(r\) stands for the radius and \(\theta\) is the central angle in radians. Mathematically, the arc length is also a function of the sector's radius. If you increase the radius while keeping the central angle constant, the arc gets longer. Knowing how to compute the arc length is essential for deriving other properties of the sector, such as the area.

Radius: The Building Block of Circular Measurements

The radius is one of the most fundamental concepts in geometry related to circles and circular sectors. It's the straight-line distance from the very center of the circle to any point on its circumference. It is effectively the 'half-width' of the circle and plays a vital role not just in defining the size of the circle but also in various calculations involving circles.

When we speak about the area of a circular sector, the radius is squared, as seen in the formula \(area = \frac{r^2\theta}{2}\). This implies that the area of the sector grows quadratically with respect to the radius; if you double the radius, the area of the sector becomes four times larger, holding the central angle constant. The radius is not just a measurement; it's a powerful influencer in the geometry of circles, drastically affecting values such as circumference, area, and in the context of sectors, the arc length.