Enter at least 3 values (including at least one side) to solve the triangle.
In the vast landscape of mathematics, few shapes are as fundamental or as versatile as the Triangle. As the simplest possible polygon—requiring only three sides and three angles to exist—it forms the structural backbone of everything from modern architecture to the complex navigation systems that guide our aircraft. Understanding the relationships between its parts is not just an academic exercise; it is a critical skill for engineers, designers, and students alike.
Our free Triangle Solver (located above) is a high-precision digital tool designed to help you solve any valid triangle combination. Whether you are working with Side-Side-Side (SSS), Side-Angle-Side (SAS), or Angle-Side-Angle (ASA), this calculator provides instant results for areas, perimeters, and missing dimensions. This comprehensive guide will take you on a deep dive into the laws of trigonometry, Heron’s formula, and the practical applications of triangle math in the real world.
A triangle is defined by six primary parts: three sides (a, b, c) and three internal angles (A, B, C). In standard mathematical notation, angle A is always opposite side a, angle B is opposite side b, and angle C is opposite side c.
The most important rule in Euclidean geometry regarding triangles is that the interior angles always sum to exactly 180 degrees.
$$A + B + C = 180^\circ$$This rule is the first check performed by our calculator. If you enter two angles that sum to 180 or more, a triangle cannot exist in a flat plane.
Before performing calculations, it is helpful to categorize your triangle. Triangles are classified in two ways: by their side lengths and by their internal angles.
Equilateral: All three sides are equal, and all three angles are exactly 60 degrees.
Isosceles: At least two sides are equal, and the angles opposite those sides are also equal.
Scalene: All sides and all angles are different.
Acute: All three internal angles are less than 90 degrees.
Right: Exactly one angle is exactly 90 degrees. These triangles follow the Pythagorean Theorem.
Obtuse: Exactly one angle is greater than 90 degrees.
Solving a triangle means finding all six parts when only three are known. Depending on what information you provide, our calculator uses different mathematical strategies.
If one angle is 90 degrees, the relationship between the sides is governed by the most famous formula in math:
$$a^2 + b^2 = c^2$$Where c is the hypotenuse (the side opposite the right angle).
When you know two sides and the angle between them (SAS) or all three sides (SSS), the Law of Cosines is the primary tool. It is essentially a generalized version of the Pythagorean Theorem that works for any triangle.
$$c^2 = a^2 + b^2 – 2ab \cos(C)$$This formula allows us to “bridge” the gap between side lengths and angle measures with extreme precision.
When you know two angles and a side (ASA or AAS), the Law of Sines provides a simple ratio that links the sides and angles.
$$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$$This is the most efficient way to calculate lengths when the triangle is “stretched” or non-right.
In elementary school, we learn that the area of a triangle is 1/2 * base * height. However, in the real world, we rarely know the “height” (the perpendicular altitude). Our calculator uses more sophisticated methods.
If you know two sides and the included angle, the area (K) can be found using:
$$K = \frac{1}{2} ab \sin(C)$$If you only know the three side lengths, the calculator uses Heron’s Formula. First, it calculates the Semi-perimeter (s):
$$s = \frac{a + b + c}{2}$$Then, it finds the area using:
$$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$$This formula is incredibly powerful because it requires zero knowledge of angles to provide a perfect area result.
Triangles are the only polygons that are inherently rigid. If you have four pieces of wood joined at the corners (a square), you can push it into a diamond shape. But if you have three pieces of wood (a triangle), the shape cannot be changed without breaking the wood or the joints. This property makes them the king of engineering.
Look at any bridge or crane, and you will see a series of triangles (trusses). Engineers use triangle calculations to distribute weight and stress evenly across a structure, ensuring stability in wind and earthquakes.
Global Positioning Systems use a process called Trilateration. By calculating the distance between your phone and three different satellites, the system forms a series of triangles to pinpoint your exact coordinates on Earth.
Every 3D model you see in a video game or a Pixar movie is actually made of thousands (or millions) of tiny triangles called “polygons.” Modern graphics cards are specialized to perform triangle calculations billions of times per second to render realistic shadows and movement.
When solving triangles, there is one scenario that can be tricky: Side-Side-Angle (SSA).
If the side opposite the known angle is shorter than the other known side, there might be two possible triangles that fit the description (one acute and one obtuse), or there might be zero possible triangles if the side is too short to reach.
Our calculator will alert you if the values you enter are geometrically impossible.
Check Your Units: Ensure all sides are in the same unit (meters, inches, etc.) before calculating.
Deg vs Rad: Most real-world applications use Degrees, but calculus and advanced physics often use Radians. Our calculator defaults to degrees for ease of use.
The Triangle Inequality Theorem: Remember that the sum of any two sides must always be greater than the third side. a + b > c. If this isn’t true, the sides can’t connect!
The Triangle Calculator is more than a homework helper; it is a gateway to understanding the physical world. It bridges the gap between raw measurements and functional design. By mastering the laws of sines and cosines and understanding the classification of these shapes, you gain a deeper appreciation for the structures and systems that surround us.
Whether you are a carpenter cutting a roof rafter, a student studying for the SAT, or a hobbyist designer, use this tool to ensure your work is mathematically sound. Mathematics is the blueprint of reality, and the triangle is its most sturdy block.
Start solving your geometry problems today. Input your dimensions, and let the Triangle Solver reveal the hidden angles and areas of your project.