Triangle Calculator

What is a Triangle Calculator?

A triangle calculator is a tool that helps compute various properties and measurements of triangles. It can calculate side lengths, angles, area, perimeter, and other geometric characteristics based on given inputs.

Key Features of Triangle Calculators

• Solve for missing sides or angles
• Calculate area and perimeter
• Determine triangle type (e.g., scalene, isosceles, equilateral)
• Compute special line segments (e.g., medians, altitudes, angle bisectors)
• Visualize the triangle based on input values

Formulas for Triangles

Let \(a\), \(b\), and \(c\) be the lengths of the sides of a triangle, and \(A\), \(B\), and \(C\) be the angles opposite to these sides respectively. Then:

1. Perimeter: \(P = a + b + c\)
2. Area (Heron's formula): \(A = \sqrt{s(s-a)(s-b)(s-c)}\), where \(s = \frac{a+b+c}{2}\) (semi-perimeter)
3. Angles (Law of Cosines): \[A = \arccos(\frac{b^2 + c^2 - a^2}{2bc})\] \[B = \arccos(\frac{a^2 + c^2 - b^2}{2ac})\] \[C = \arccos(\frac{a^2 + b^2 - c^2}{2ab})\]
4. Heights: \[h_a = \frac{2A}{a}, h_b = \frac{2A}{b}, h_c = \frac{2A}{c}\]
5. Angle Bisectors: \[t_a = \frac{2bc\cos(A/2)}{b+c}, t_b = \frac{2ac\cos(B/2)}{a+c}, t_c = \frac{2ab\cos(C/2)}{a+b}\]
6. Medians: \[m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}\] \[m_b = \frac{1}{2}\sqrt{2a^2 + 2c^2 - b^2}\] \[m_c = \frac{1}{2}\sqrt{2a^2 + 2b^2 - c^2}\]

Example Calculations

Let's calculate these properties for a triangle with sides \(a = 3\), \(b = 4\), and \(c = 5\) units:

1. Perimeter: \[P = a + b + c = 3 + 4 + 5 = 12 \text{ units}\]
2. Semi-perimeter: \[s = \frac{a+b+c}{2} = \frac{12}{2} = 6 \text{ units}\]
3. Area: \[A = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{6(6-3)(6-4)(6-5)} = \sqrt{6 \cdot 3 \cdot 2 \cdot 1} = \sqrt{36} = 6 \text{ square units}\]
4. Angles: \begin{align*} A &= \arccos(\frac{4^2 + 5^2 - 3^2}{2 \cdot 4 \cdot 5}) \approx 36.87^\circ \\ B &= \arccos(\frac{3^2 + 5^2 - 4^2}{2 \cdot 3 \cdot 5}) \approx 53.13^\circ \\ C &= \arccos(\frac{3^2 + 4^2 - 5^2}{2 \cdot 3 \cdot 4}) = 90^\circ \end{align*}
5. Heights: \begin{align*} h_a &= \frac{2A}{a} = \frac{2 \cdot 6}{3} = 4 \text{ units} \\ h_b &= \frac{2A}{b} = \frac{2 \cdot 6}{4} = 3 \text{ units} \\ h_c &= \frac{2A}{c} = \frac{2 \cdot 6}{5} = 2.4 \text{ units} \end{align*}
6. Angle Bisectors: \begin{align*} t_a &= \frac{2bc\cos(A/2)}{b+c} = \frac{2 \cdot 4 \cdot 5 \cdot \cos(36.87^\circ/2)}{4+5} \approx 3.61 \text{ units} \\ t_b &= \frac{2ac\cos(B/2)}{a+c} = \frac{2 \cdot 3 \cdot 5 \cdot \cos(53.13^\circ/2)}{3+5} \approx 3.28 \text{ units} \\ t_c &= \frac{2ab\cos(C/2)}{a+b} = \frac{2 \cdot 3 \cdot 4 \cdot \cos(90^\circ/2)}{3+4} \approx 2.40 \text{ units} \end{align*}
7. Medians: \begin{align*} m_a &= \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2} = \frac{1}{2}\sqrt{2 \cdot 4^2 + 2 \cdot 5^2 - 3^2} \approx 4.27 \text{ units} \\ m_b &= \frac{1}{2}\sqrt{2a^2 + 2c^2 - b^2} = \frac{1}{2}\sqrt{2 \cdot 3^2 + 2 \cdot 5^2 - 4^2} \approx 3.61 \text{ units} \\ m_c &= \frac{1}{2}\sqrt{2a^2 + 2b^2 - c^2} = \frac{1}{2}\sqrt{2 \cdot 3^2 + 2 \cdot 4^2 - 5^2} = 3 \text{ units} \end{align*}

Visual Representation

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