What Are Inscribed Angles and Why Should You Care?
An inscribed angle is an angle formed by two chords in a circle that intersect at a single point on the circle's circumference. The noteworthy part? The vertex of this angle is on the circle itself, making it a key player in various geometric properties and theorems.
Whether you're into solving geometry problems, visualizing beautiful diagrams, or simply looking to ace your next math test, understanding inscribed angles can be super helpful.
How to Calculate Inscribed Angles
To calculate the inscribed angle, you'll need two key pieces of information:
- The length of the minor arc
- The radius of the circle
The formula to compute the inscribed angle uses these values:
[A = \left( \frac{\text{Arc Length}}{2 \pi \times \text{Radius}} \right) \times \frac{360}{2}]
Where:
- Arc Length is the length of the minor arc
- Radius is the radius of the circle
This formula derives from the relationship between the minor arc and the central angle, divided by two since the inscribed angle is half the central angle intercepted by the same arc.
Calculation Example
Imagine you're given the following:
- The minor arc length is 10 units
- The radius of the circle is 15 units
Let's plug these values into the formula:
[A = \left( \frac{10}{2 \pi \times 15} \right) \times \frac{360}{2}]
First, compute the part inside the parentheses:
[\frac{10}{2 \pi \times 15} = \frac{10}{30 \pi} \approx \frac{10}{94.248} \approx 0.106]
Next, multiply by 180:
[A = 0.106 \times 180 \approx 19.08°]
So, the inscribed angle is approximately 19.08 degrees.