# Triangle Length Calculator

## Triangle Length Calculator

The Triangle Length Calculator is a very useful tool that can help you quickly determine, just enter the lengths of the triangle's sides (Side A, Side B, and Side C) and press enter to findout

### What is Triangle Length?

The length of a triangle refers to the lengths of its sides. A triangle consists of three sides, often labeled as Side A, Side B, and Side C. Knowing the lengths of these sides, you can calculate the internal angles of the triangle.

### Calculation Examples

#### Example 1

**Inputs:**

- Side A: 7 cm
- Side B: 9 cm
- Side C: 12 cm

**Step-by-Step Calculation:**

**1. Calculate Angle α (alpha):**

$$\alpha ={\mathrm{cos}}^{-1}\left(\frac{{b}^{2}+{c}^{2}-{a}^{2}}{2bc}\right)$$

$$\alpha ={\mathrm{cos}}^{-1}\left(\frac{{9}^{2}+1{2}^{2}-{7}^{2}}{2\cdot 9\cdot 12}\right)$$

$$\alpha ={\mathrm{cos}}^{-1}\left(\frac{81+144-49}{216}\right)$$

$$\alpha ={\mathrm{cos}}^{-1}\left(\frac{176}{216}\right)$$

$\alpha = \cos^{-1}(0.8148) \approx 35.3^\circ$

**2. Calculate Angle β (beta):**

$$\beta ={\mathrm{cos}}^{-1}\left(\frac{{a}^{2}+{c}^{2}-{b}^{2}}{2ac}\right)$$

$$\beta ={\mathrm{cos}}^{-1}\left(\frac{{7}^{2}+1{2}^{2}-{9}^{2}}{2\cdot 7\cdot 12}\right)$$

$$\beta ={\mathrm{cos}}^{-1}\left(\frac{49+144-81}{168}\right)$$

$$\beta ={\mathrm{cos}}^{-1}\left(\frac{112}{168}\right)$$

$\beta = \cos^{-1}(0.6667) \approx 48.2^\circ$

**3. Calculate Angle γ (gamma):**

$$\gamma =18{0}^{\circ}-\alpha -\beta $$

$\gamma = 180^\circ - 35.3^\circ - 48.2^\circ \approx 96.5^\circ$

**3. Results:**

- Angle α: 35.3° (deg)
- Angle β: 48.2° (deg)
- Angle γ: 96.5° (deg)

#### Example 2

**Inputs:**

- Side A: 5 ft
- Side B: 6 ft
- Side C: 7 ft

**Step-by-Step Calculation:**

**1. Calculate Angle α (alpha):**

$$\alpha ={\mathrm{cos}}^{-1}\left(\frac{{b}^{2}+{c}^{2}-{a}^{2}}{2bc}\right)$$

$$\alpha ={\mathrm{cos}}^{-1}\left(\frac{{6}^{2}+{7}^{2}-{5}^{2}}{2\cdot 6\cdot 7}\right)$$

$$\alpha ={\mathrm{cos}}^{-1}\left(\frac{36+49-25}{84}\right)$$

$$\alpha ={\mathrm{cos}}^{-1}\left(\frac{60}{84}\right)$$

$\alpha = \cos^{-1}(0.7143) \approx 44.4^\circ$

**2. Calculate Angle β (beta):**

$$\beta ={\mathrm{cos}}^{-1}\left(\frac{{a}^{2}+{c}^{2}-{b}^{2}}{2ac}\right)$$

$$\beta ={\mathrm{cos}}^{-1}\left(\frac{{5}^{2}+{7}^{2}-{6}^{2}}{2\cdot 5\cdot 7}\right)$$

$$\beta ={\mathrm{cos}}^{-1}\left(\frac{25+49-36}{70}\right)$$

$$\beta ={\mathrm{cos}}^{-1}\left(\frac{38}{70}\right)$$

$\beta = \cos^{-1}(0.5429) \approx 57.1^\circ$

**2. Calculate Angle γ (gamma):**

$$\gamma =18{0}^{\circ}-\alpha -\beta $$

$\gamma = 180^\circ - 44.4^\circ - 57.1^\circ \approx 78.5^\circ$

**3. Results:**

- Angle α: 44.4° (deg)
- Angle β: 57.1° (deg)
- Angle γ: 78.5° (deg)