Problem 57 - Algebra and Trigonometry

Question

Find the radius of the circle if an arc of length 6 \(\mathrm{m}\) on the circle subtends a central angle of \(\pi / 6 \mathrm{rad} .\)

Step-by-Step Solution

Answer

The radius of the circle is \( \frac{36}{\pi} \) meters.

Step 1: Identify the Given Information

We are given the length of an arc, which is 6 meters, and the central angle subtended by this arc is \(\pi/6\) radians. Our task is to find the radius of the circle.

Step 2: Recall the Formula for Arc Length

The formula for the length of an arc \( L \) of a circle, given the central angle \( \theta \) in radians and the radius \( r \), is \[ L = r \theta \]

Step 3: Substitute Known Values into Formula

Substitute the given values into the formula: the arc length \( L = 6 \) meters and the angle \( \theta = \pi/6 \). This gives us the equation \[ 6 = r \left( \frac{\pi}{6} \right) \]

Step 4: Solve for the Radius

To find \( r \), solve the equation from Step 3: \[ 6 = \frac{r \pi}{6} \] Multiply both sides by 6 to eliminate the fraction: \[ 36 = r \pi \] Now divide both sides by \( \pi \) to solve for \( r \): \[ r = \frac{36}{\pi} \]

Key Concepts

Arc Length

The concept of arc length is a fundamental aspect of circle geometry. An arc is a part of the circumference of a circle. The **arc length** is simply the measure of this curved path.

To find the arc length, we use the formula:

• \[ L = r \theta\]

This formula indicates that the arc length (\(L\)) is the product of the radius (\(r\)) of the circle and the central angle (\(\theta\)) that it subtends, measured in radians.

For example, if you have an arc length of 6 meters and the central angle is \(\pi/6\) radians, you'd plug these values into the equation to find the radius. This relationship highlights the connection between the portion of the circle and the angle at the center formed by the endpoints of the arc.

Central Angle

Understanding the central angle is crucial in circle geometry. The **central angle** is the angle formed at the center of the circle by two radii. It measures how "wide" an arc opens up from the center, essentially telling you how big the slice of the pie is.

Central angles are measured in radians, providing a straightforward relationship between the angle and the arc length.

A full circle is \(2\pi\) radians, which means a central angle's measure in radians can be a fraction of \(2\pi\).

For example, in our exercise where the central angle is \(\pi/6\) radians, this angle represents one-twelfth of a full circle, corresponding to a specific arc length on the circle's circumference. This relationship shows why radians are often preferred over degrees when dealing with arc length and circle geometry, due to their mathematical convenience.

Circle Geometry

Circle geometry is an engaging area of mathematics focused on the properties and equations governing circles.

Circles are defined by their radius, the distance from the center to any point on the circumference. The larger the radius, the larger the circle.

When dealing with problems in circle geometry, such as finding the radius from an arc length and a central angle, a few key concepts always come into play:

• The **radius** is crucial in calculating properties of the circle, like the circumference and area.
• The relationship between the arc length and the central angle simplifies this type of problem, as seen through direct applications of known formulas.

In essence, circle geometry encompasses understanding how these elements interact to form various properties and measurements of a circle.