Triangle Calculator

Solving a triangle means finding all its unknown sides and angles when some of them are already known. To do this, our calculator uses standard geometric rules and trigonometric identities. Depending on which inputs you provide, the algorithm categorizes your problem into one of five standard cases: SSS, SAS, ASA, AAS, or SSA.

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The Trigonometric Laws Applied

Our engine utilizes three main mathematical relationships to solve oblique and right triangles.

1. Angle Sum Theorem

In Euclidean geometry, the interior angles of any flat triangle always sum to exactly $180^\circ$ (or $\pi$ radians):

$$A + B + C = 180^\circ$$

2. The Law of Sines

The Law of Sines establishes that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides of a triangle:

$$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$$

This is ideal for solving triangles when you know an angle and its opposite side along with one other piece of information (e.g., ASA, AAS, or SSA).

3. The Law of Cosines

The Law of Cosines is a generalization of the Pythagorean theorem that relates the lengths of the sides to the cosine of one of its angles:

$$a^2 = b^2 + c^2 - 2bc \cos(A)$$ $$b^2 = a^2 + c^2 - 2ac \cos(B)$$ $$c^2 = a^2 + b^2 - 2ab \cos(C)$$

This is used primarily when solving SSS (three sides) or SAS (two sides and their included angle) scenarios.

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Step-by-Step Breakdown of Input Scenarios

Case 1: Side-Side-Side (SSS)

When all three side lengths $a$, $b$, and $c$ are known:

1. We check if they form a valid triangle using the triangle inequality theorem ($a + b > c$).
2. We isolate the angles using the rearranged Law of Cosines: $$A = \arccos\left(\frac{b^2 + c^2 - a^2}{2bc}\right)$$ $$B = \arccos\left(\frac{a^2 + c^2 - b^2}{2ac}\right)$$
3. The remaining angle is found with $C = 180^\circ - (A + B)$.

Case 2: Side-Angle-Side (SAS)

When you know two sides (e.g., $a$ and $b$) and the angle between them ($C$):

1. Find the unknown side $c$ using the Law of Cosines: $$c = \sqrt{a^2 + b^2 - 2ab \cos(C)}$$
2. Find angle $A$ using the Law of Cosines (or Sines): $$A = \arccos\left(\frac{b^2 + c^2 - a^2}{2bc}\right)$$
3. Find the final angle: $B = 180^\circ - (A + C)$.

Case 3: Angle-Side-Angle (ASA) & Angle-Angle-Side (AAS)

When two angles and one side are given:

1. Instantly calculate the third angle using the Angle Sum Theorem: $$C = 180^\circ - (A + B)$$
2. Use the Law of Sines to find the two remaining unknown sides: $$a = c \cdot \frac{\sin(A)}{\sin(C)}$$ $$b = c \cdot \frac{\sin(B)}{\sin(C)}$$

Case 4: Side-Side-Angle (SSA) — The Ambiguous Case

When two sides ($a, b$) and a non-included angle ($A$) are provided, the system must navigate the ambiguous case. Depending on the lengths and the angle, this scenario can result in:

• Zero Triangles: If the side $a$ is too short to reach the base line (i.e., $a < b \sin(A)$ when $A$ is acute).
• One Right Triangle: If $a = b \sin(A)$, forming a perfect right triangle.
• Two Unique Triangles: If $b \sin(A) < a < b$, two distinct sets of angles and sides can satisfy the input. The calculator will provide both valid solutions.
• One Oblique Triangle: If $a \geq b$.

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Calculating Area, Perimeter, and Altitudes

Once all three sides ($a, b, c$) are computed:

• Perimeter: $P = a + b + c$
• Semi-perimeter: $s = \frac{a+b+c}{2}$
• Area (Heron's Formula): $$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$$
• Altitudes: The altitude (height) perpendicular to each base is computed using the standard area equation ($\text{Area} = \frac{1}{2} \cdot \text{base} \cdot \text{height}$): $$h_a = \frac{2 \cdot \text{Area}}{a}, \quad h_b = \frac{2 \cdot \text{Area}}{b}, \quad h_c = \frac{2 \cdot \text{Area}}{c}$$