Problem 39 - Algebra 2 and Trigonometry

Question

In a circle whose radius measures 5 feet, a central angle intercepts an are of length 12 feet. Find the radian measure of the central angle.

Step-by-Step Solution

Answer

The radian measure of the central angle is approximately 2.4 radians.

Step 1: Understanding the Problem

We are asked to find the radian measure of a central angle that intercepts an arc in a circle. We know the radius of the circle, which is 5 feet, and the arc length, which is 12 feet. The formula relating arc length (\(s\)), radius (\(r\)), and the radian measure of the central angle (\(\theta\)) is given by: \[ s = r \cdot \theta \].

Step 2: Substitute Known Values into the Formula

We substitute the known values into the formula \[ s = r \cdot \theta \]. Here, \(s = 12\) feet and \(r = 5\) feet. This gives us the equation \[ 12 = 5 \cdot \theta.\]

Step 3: Solve for the Central Angle

To find the angle \(\theta\), we need to solve the equation \[ 12 = 5 \cdot \theta \]. Divide both sides of the equation by 5 to isolate \(\theta\): \[ \theta = \frac{12}{5}. \]

Step 4: Convert Fraction to Decimal Form

Sometimes it's easier to interpret results in decimal form. Calculate \( \frac{12}{5} \), which is 2.4. Thus, the radian measure of the central angle is \(\theta \approx 2.4\) radians.

Key Concepts

Central Angle

The concept of a central angle is fundamental in understanding circles and their properties. In simple terms, a central angle is an angle whose vertex is at the center of the circle. The two rays that form the angle extend to the circle, intercepting an arc on the circle's boundary.

To measure a central angle, we often use radians instead of degrees. A radian is a measure of angle based on the radius of the circle. Unlike degrees - which divides a circle into 360 parts - radians use the circle's radius for measurement. Specifically,

• 1 radian is the angle formed when the arc length equals the radius.
• To convert an angle from degrees to radians, multiply by \(\frac{\pi}{180}\).
• Radian is the standard unit of angular measure used in many areas of mathematics.

A central angle provides critical information related to the circle's geometry, like the arc length or sectors formed within the circle.

Arc Length

The arc length is a portion of the circle's circumference, determined by the central angle. When an angle intercepts an arc, it delineates how much of the circle's perimeter belongs to that arc. The arc length formula is

• \(s = r \cdot \theta \)

where \(s\) is the arc length, \(r\) is the radius, and \(\theta\) is the central angle in radians.

This formula is valuable because it provides a straightforward method for calculating the arc length based on simple inputs like radius and angle. The arc length measurement is important for understanding proportions of the circle, both in theory and in practical applications, like engineering or construction.

• If \(\theta\) is in degrees, you must convert it to radians first.
• Arc length is essentially a piece of the circle - think of it as part of the circle's crust if the circle were a pie.

Circle Radius

The circle's radius is a key measurement originating from the circle's center to its boundary. The radius plays a pivotal role in defining the circle's size and its associated parameters, such as area and circumference.

• The formula for the circumference of a circle is \(C = 2 \pi r\).
• The area of a circle is \(A = \pi r^2\).

For a circle formed by a central angle, understanding the radius is crucial as it is directly used in calculating the arc length.

Since a basic principle is that all radii in a circle are equal, this measurement serves as a reliable foundation for calculations pertaining to the circle.

• When discussing angular measurements like radians, the radius helps establish a common reference point.
• For practical purposes, the radius can be measured using a string and a ruler, making it an accessible and fundamental geometry tool.

Algebraic Equations

Algebraic equations are the backbone of solving mathematical problems involving geometric shapes like circles. In the context of determining the radian measure of a central angle, algebra simplifies the process of calculation.

In our exercise, we use the arc length formula:

• \(12 = 5 \cdot \theta\)
• Solving it involves isolating \(\theta\) by dividing both sides by 5.

This highlights how algebra enables us to express and solve geometric problems with ease. The beauty of algebra is that it provides:

• A symbolic way to represent problems where unknown variables are to be determined.
• A systemic approach to solving for these variables through operations such as addition, subtraction, multiplication, and division.

By breaking down geometric complexities into straightforward algebraic expressions, calculations become simpler and more efficient, helping students grasp concepts better.