Data Source & Methodology

This calculator uses standard trigonometric and geometric principles for its calculations. All results are derived from foundational mathematical laws, ensuring precision and reliability.

• AuthoritativeDataSource: Law of Cosines, Law of Sines, and Heron's Formula.
• Reference: These are foundational principles of trigonometry and geometry, universally accepted and taught in standard mathematics curricula. See, for example, "subcategories/geometry" (Moise & Downs, 1991) or any standard university-level trigonometry text.

All calculations are based strictly on these mathematical laws. Angles are returned in degrees.

The Formulas Explained

The formulas used depend on the data you provide (SSS, SAS, or ASA).

1. Given 3 Sides (SSS)

When sides $a$, $b$, and $c$ are known:

1. Semi-Perimeter ($s$): First, we calculate the semi-perimeter.
   $ s = \frac{a + b + c}{2} $
2. Area (Heron's Formula): The area is found using Heron's Formula.
   $ Area = \sqrt{s(s-a)(s-b)(s-c)} $
3. Angles (Law of Cosines): The angles are found using the Law of Cosines.
   $ \alpha = \arccos\left(\frac{b^2 + c^2 - a^2}{2bc}\right) $
   $ \beta = \arccos\left(\frac{a^2 + c^2 - b^2}{2ac}\right) $
   $ \gamma = \arccos\left(\frac{a^2 + b^2 - c^2}{2ab}\right) $

2. Given 2 Sides & 1 Included Angle (SAS)

When sides $a$, $b$, and the included angle $\gamma$ are known:

1. Third Side (Law of Cosines): We find the third side, $c$.
   $ c = \sqrt{a^2 + b^2 - 2ab \cos(\gamma)} $
2. Area: The area can be calculated directly.
   $ Area = \frac{1}{2}ab \sin(\gamma) $
3. Other Angles (Law of Sines): The remaining angles are found using the Law of Sines.
   $ \alpha = \arcsin\left(\frac{a \sin(\gamma)}{c}\right) $
   $ \beta = 180^\circ - \alpha - \gamma $

3. Given 2 Angles & 1 Included Side (ASA)

When angles $\alpha$, $\beta$, and the included side $c$ are known:

1. Third Angle: The third angle is simple.
   $ \gamma = 180^\circ - \alpha - \beta $
2. Other Sides (Law of Sines): We find the other sides using the Law of Sines.
   $ a = \frac{c \sin(\alpha)}{\sin(\gamma)} $
   $ b = \frac{c \sin(\beta)}{\sin(\gamma)} $
3. Area: With all sides, we can use Heron's formula, or this variant:
   $ Area = \frac{c^2 \sin(\alpha) \sin(\beta)}{2 \sin(\gamma)} $

All Cases: Heights and Perimeter

• Perimeter: $ P = a + b + c $
• Heights: The height (altitude) $h_a$ from vertex A to side $a$ is:
  $ h_a = \frac{2 \cdot Area}{a} $
  (This is repeated for $h_b$ and $h_c$).

Glossary of Variables

• $a, b, c$: The lengths of the three sides of the triangle.
• $\alpha, \beta, \gamma$: The three interior angles (in degrees). Angle $\alpha$ is opposite side $a$, $\beta$ is opposite $b$, and $\gamma$ is opposite $c$.
• Area: The total surface area enclosed by the triangle.
• Perimeter: The total length of the triangle's boundary ($a+b+c$).
• $h_a, h_b, h_c$: The heights (altitudes) of the triangle from a vertex perpendicular to the opposite side ($a, b, c$ respectively).

How it Works: A Step-by-Step Example

Let's calculate the properties of a scalene triangle with sides $a = 5$, $b = 6$, and $c = 7$.

1. Select Mode: Choose "3 Sides (SSS)" from the dropdown.
2. Enter Inputs:
   • Set Side $a$ = 5
   • Set Side $b$ = 6
   • Set Side $c$ = 7
3. Calculate: Click the "Calculate" button.
4. Review Results: The calculator first validates the input (Triangle Inequality: 5+6 > 7, 5+7 > 6, 6+7 > 5. It's valid.)
5. Formulas Applied:
   • Semi-Perimeter ($s$): $ (5 + 6 + 7) / 2 = 9 $
   • Area (Heron's): $ \sqrt{9(9-5)(9-6)(9-7)} = \sqrt{9 \cdot 4 \cdot 3 \cdot 2} = \sqrt{216} \approx 14.697 $
   • Angle $\alpha$ (Law of Cosines): $ \arccos\left(\frac{6^2 + 7^2 - 5^2}{2 \cdot 6 \cdot 7}\right) = \arccos(0.714...) \approx 44.415^\circ $
   • Angle $\beta$: $ \approx 57.122^\circ $
   • Angle $\gamma$: $ \approx 78.463^\circ $
   • Height $h_a$: $ (2 \cdot 14.697) / 5 \approx 5.879 $

Frequently Asked Questions (FAQ)

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$ s = \frac{a + b + c}{2} $

$ Area = \sqrt{s(s-a)(s-b)(s-c)} $

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

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

$ \gamma = \arccos\left(\frac{a^2 + b^2 - c^2}{2ab}\right) $

$ c = \sqrt{a^2 + b^2 - 2ab \cos(\gamma)} $