Circumradius of Scalene Triangle Calculator

Circumradius Formula:

\[ r_c = \frac{S_{\text{Shorter}}}{2 \times \sin(\angle_{\text{Smaller}})} \]

Circumradius of Scalene Triangle:

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1. What is the Circumradius of a Scalene Triangle?

The circumradius of a scalene triangle is the radius of the circumscribed circle that passes through all three vertices of the triangle. It's a fundamental geometric property that helps characterize the triangle's size and shape.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{S_{\text{Shorter}}}{2 \times \sin(\angle_{\text{Smaller}})} \]

Where:

\( r_c \) — Circumradius of the scalene triangle
\( S_{\text{Shorter}} \) — Length of the shorter side
\( \angle_{\text{Smaller}} \) — Measure of the smaller angle (in radians)

Explanation: This formula relates the circumradius to the shortest side and its opposite angle using the sine function from trigonometry.

3. Importance of Circumradius Calculation

Details: Calculating the circumradius is important in various geometric applications, including triangle analysis, circle packing problems, and in engineering designs involving triangular structures.

4. Using the Calculator

Tips: Enter the length of the shorter side in meters and the smaller angle in degrees. The angle must be between 0° and 180° (exclusive) for a valid triangle.

5. Frequently Asked Questions (FAQ)

Q1: Why use the shorter side and smaller angle specifically?
A: This formula provides the most direct calculation when you know the shortest side and its opposite angle, which is often easier to measure in practical applications.

Q2: Can I use this formula with any side and opposite angle?
A: Yes, the formula works for any side and its opposite angle, but it's most accurate when using the shortest side and smallest angle due to measurement precision.

Q3: What units should I use for the inputs?
A: The side length can be in any unit (the result will be in the same unit), and the angle should be in degrees.

Q4: Does this work for all types of triangles?
A: This specific formula works for scalene triangles, but similar principles apply to isosceles and equilateral triangles with appropriate modifications.

Q5: What if my angle is exactly 0° or 180°?
A: These values would not form a valid triangle. The angle must be strictly between 0° and 180°.