Isosceles Triangle Angles Calculator With Sides

Isosceles Triangle Angles Formula:

\[ \text{Vertex Angle} = \arccos\left(\frac{\text{leg}^2 + \text{leg}^2 - \text{base}^2}{2 \times \text{leg} \times \text{leg}}\right) \] \[ \text{Base Angle} = \frac{180^\circ - \text{Vertex Angle}}{2} \]

Vertex Angle:

Base Angle:

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1. What is an Isosceles Triangle?

An isosceles triangle is a triangle with two sides of equal length and two angles of equal measure. The angles opposite the equal sides are equal, and the angle between the equal sides is called the vertex angle.

2. How Does the Calculator Work?

The calculator uses the Law of Cosines to find the vertex angle:

\[ \text{Vertex Angle} = \arccos\left(\frac{\text{leg}^2 + \text{leg}^2 - \text{base}^2}{2 \times \text{leg} \times \text{leg}}\right) \]

Then calculates the base angles:

\[ \text{Base Angle} = \frac{180^\circ - \text{Vertex Angle}}{2} \]

Where:

\( \text{leg} \) — Length of the equal sides
\( \text{base} \) — Length of the unequal side
\( \arccos \) — Inverse cosine function

3. Importance of Angle Calculation

Details: Calculating angles in isosceles triangles is fundamental in geometry, architecture, engineering, and various design applications where symmetry is important.

4. Using the Calculator

Tips: Enter the lengths of the equal sides and the base in any consistent units. Select whether you want results in degrees or radians. All lengths must be positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: What's the range for vertex angle in an isosceles triangle?
A: The vertex angle can range from just above 0° to just below 180° (or 0 to π radians).

Q2: Can this calculator be used for equilateral triangles?
A: Yes, when all sides are equal, it will correctly calculate all angles as 60° (or π/3 radians).

Q3: What if I know the angles and need side lengths?
A: You would need a different calculator that uses the Law of Sines or Cosines in reverse.

Q4: How precise are the calculations?
A: The calculator provides results with 2 decimal places for degrees and 4 for radians, though internal calculations use higher precision.

Q5: What about degenerate triangles?
A: The calculator validates inputs to ensure they can form a valid triangle (sum of two sides must be greater than the third).