Isosceles Triangle Calculator

What is an Isosceles Triangle?

An isosceles triangle is a triangle with at least two equal sides. The word "isosceles" comes from the Greek words "isos" (equal) and "skelos" (leg). In an isosceles triangle, the angles opposite the two equal sides are also equal.

Key Properties of Isosceles Triangles

• Two sides of equal length (legs)
• Two angles of equal measure (base angles)
• An axis of symmetry through the vertex angle

Formulas for Isosceles Triangles

Let \(a\) be the length of each equal side (leg), and \(b\) be the length of the base. Then:

1. Perimeter: \(P = 2a + b\)
2. Area: \(A = \frac{1}{4}b\sqrt{4a^2 - b^2}\)
3. Height (to base): \(h = \sqrt{a^2 - (\frac{b}{2})^2}\)
4. Base angles: \(\theta = \arccos(\frac{b}{2a})\)
5. Vertex angle: \(\phi = 180^\circ - 2\theta\)

Step-by-Step Calculations

Let's calculate these properties for an isosceles triangle with legs \(a = 5\) units and base \(b = 6\) units:

1. Perimeter: \[P = 2a + b = 2(5) + 6 = 16 \text{ units}\]
2. Area: \[A = \frac{1}{4}b\sqrt{4a^2 - b^2} = \frac{1}{4}(6)\sqrt{4(5^2) - 6^2} = \frac{3}{2}\sqrt{64} = 12 \text{ square units}\]
3. Height: \[h = \sqrt{a^2 - (\frac{b}{2})^2} = \sqrt{5^2 - 3^2} = 4 \text{ units}\]
4. Base angles: \[\theta = \arccos(\frac{b}{2a}) = \arccos(\frac{6}{2(5)}) \approx 53.13^\circ\]
5. Vertex angle: \[\phi = 180^\circ - 2\theta \approx 180^\circ - 2(53.13^\circ) \approx 73.74^\circ\]

Visual Representation

This diagram illustrates the isosceles triangle with the calculated dimensions and angles.