Isosceles Triangle Calculator

How to Calculate Isosceles Triangle Properties

An isosceles triangle is a triangle with two equal sides and two equal angles. Understanding its properties is crucial in geometry and various real-world applications. Here's how to calculate different aspects of an isosceles triangle:

Formulas for Isosceles Triangle Calculations

Let \( a \) be the length of the two equal sides, \( b \) be the length of the base, \( h \) be the height, and \( \theta \) be the apex angle.

• Height: \( h = \sqrt{a^2 - (\frac{b}{2})^2} \)
• Apex Angle: \( \theta = 2 \arcsin(\frac{b}{2a}) \)
• Base Angle: \( \beta = \frac{180° - \theta}{2} \)
• Perimeter: \( P = 2a + b \)
• Area: \( A = \frac{1}{2}bh \)
• Inradius: \( r = \frac{A}{a + \frac{b}{2}} \)
• Circumradius: \( R = \frac{a}{\sin(\frac{\theta}{2})} \)

Calculation Steps

1. Identify the known values (at least two) among side length, base length, height, or apex angle.
2. Use the appropriate formulas to calculate the unknown values.
3. Calculate the perimeter by adding the two equal sides and the base.
4. Determine the area using the base and height.
5. Compute the inradius and circumradius using their respective formulas.

Example Calculation

Let's calculate the properties of an isosceles triangle with side length \( a = 10 \) units and base length \( b = 12 \) units.

1. Height: \( h = \sqrt{10^2 - (\frac{12}{2})^2} = \sqrt{100 - 36} = \sqrt{64} = 8 \) units
2. Apex Angle: \( \theta = 2 \arcsin(\frac{12}{2 * 10}) = 2 \arcsin(0.6) \approx 73.74° \)
3. Base Angle: \( \beta = \frac{180° - 73.74°}{2} \approx 53.13° \)
4. Perimeter: \( P = 2 * 10 + 12 = 32 \) units
5. Area: \( A = \frac{1}{2} * 12 * 8 = 48 \) square units
6. Inradius: \( r = \frac{48}{10 + \frac{12}{2}} = \frac{48}{16} = 3 \) units
7. Circumradius: \( R = \frac{10}{\sin(\frac{73.74°}{2})} \approx 6.24 \) units

Visual Representation

This diagram illustrates an isosceles triangle with its key components labeled: equal sides (a), base (b), height (h), apex angle (θ), and base angles (β).