Exradius Opposite to Angle A of Triangle Calculator

Formula Used:

\[ r_e(∠A) = \sqrt{\frac{\left(\frac{S_a + S_b + S_c}{2}\right) \cdot \left(\frac{S_a - S_b + S_c}{2}\right) \cdot \left(\frac{S_a + S_b - S_c}{2}\right)}{\left(\frac{S_b + S_c - S_a}{2}\right)}} \]

Exradius Opposite to ∠A:

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1. What is the Exradius Opposite to Angle A?

The exradius opposite to ∠A of a triangle is the radius of the circle formed with its center at the point of intersection of the internal angle bisector of ∠A and the external angle bisectors of the other two angles. It represents the radius of the excircle tangent to side A and the extensions of sides B and C.

2. How Does the Calculator Work?

The calculator uses the exradius formula:

Where:

\( S_a \) — Length of side A opposite to angle A (meters)
\( S_b \) — Length of side B opposite to angle B (meters)
\( S_c \) — Length of side C opposite to angle C (meters)

Explanation: The formula calculates the exradius using the semi-perimeter and various combinations of the triangle's side lengths.

3. Importance of Exradius Calculation

Details: Calculating the exradius is important in advanced geometry problems, particularly those involving triangle properties, circle geometry, and construction problems. It helps in understanding the relationships between a triangle and its excircles.

4. Using the Calculator

Tips: Enter all three side lengths in meters. All values must be positive numbers that satisfy the triangle inequality theorem (sum of any two sides must be greater than the third side).

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between inradius and exradius?
A: The inradius is the radius of the inscribed circle (incircle) that touches all three sides from inside the triangle, while the exradius is the radius of an excircle that touches one side and the extensions of the other two sides.

Q2: How many exradii does a triangle have?
A: A triangle has three exradii, each opposite to one of the three angles (∠A, ∠B, and ∠C).

Q3: When is the exradius formula undefined?
A: The formula becomes undefined when the denominator is zero or negative, which occurs when the side lengths do not form a valid triangle according to the triangle inequality theorem.

Q4: Can the exradius be larger than the triangle's sides?
A: Yes, the exradius can be larger than any of the triangle's sides, especially in acute triangles with small angles.

Q5: What are practical applications of exradius calculations?
A: Exradius calculations are used in advanced geometry, architectural design, engineering applications involving triangular structures, and mathematical problem-solving.