Triangle Calculator

Enter the sides and angles you know, and this triangle calculator solves for the rest. It works for right triangles, oblique triangles, and every shape in between. You get missing side lengths, angles, area, perimeter, and height in seconds.

Whether you have two sides and one angle, three sides, or one side and two angles, the calculator applies the correct formula automatically. No need to remember which rule to use.

How the Triangle Calculator Works

The calculator takes your known values and determines which formula fits. It selects from the Pythagorean theorem, the law of sines, the law of cosines, or basic angle sums depending on your input combination.

Here is the general process:

1. Enter at least three known values (sides or angles). One of them must be a side length.
2. The calculator identifies the valid solution method.
3. It returns all missing sides, angles, area, perimeter, and height.

Results update instantly. If the inputs describe an impossible triangle, the calculator tells you. For example, two sides shorter than the third side cannot form a closed shape.

Sides and Angles: What You Need for a Calculation

Every triangle has three sides and three angles. To solve a triangle, you need a minimum of three known values, and at least one must be a side length.

Three angles alone are not enough. Knowing all three angles tells you the shape but not the size. You need at least one side length to pin down the actual dimensions.

When you know two sides and one angle

This is one of the most common input combinations. The approach depends on where the known angle sits relative to the two known sides.

• Angle between the two sides (SAS). The law of cosines gives you the third side directly. Then the law of sines fills in the remaining angles.
• Angle opposite one of the known sides (SSA). The law of sines finds the angle opposite the other known side. Be aware this can produce two valid triangles (the ambiguous case) or none at all.

If the calculator returns two solutions, both are geometrically valid. Check which one matches your real situation.

When you know one side and two angles

Two angles immediately give you the third because all three angles of a triangle add to 180°. From there, the law of sines calculates the remaining two side lengths using the one known side as a reference.

This combination (AAS or ASA) always produces exactly one triangle. There is no ambiguous case here.

When you know three side lengths

When three sides are known, the calculator uses the law of cosines to find each angle. This works as long as the side lengths satisfy the triangle inequality: the sum of any two sides must be greater than the third side.

If you enter lengths of all three sides and they violate this rule, no triangle exists. The calculator will flag that.

Right Triangle Calculator

A right triangle has one angle fixed at 90°. That single constraint simplifies every formula and makes the right triangle calculator the fastest way to solve these problems.

Right angle and the sides of a right triangle

A right triangle has three named sides:

• Hypotenuse. The longest side, always opposite the right angle.
• Adjacent side. The side next to a given acute angle (not the hypotenuse).
• Opposite side. The side across from a given acute angle.

The right angle locks the triangle's geometry. Knowing just one side and one acute angle is enough to solve the entire triangle.

Finding the hypotenuse with the Pythagorean theorem

The Pythagorean theorem states:

a² + b² = c²

Here, a and b are the two shorter sides and c is the hypotenuse. If you know any two sides, you solve for the third:

• Hypotenuse: c = √(a² + b²)
• A shorter side: a = √(c² − b²)

This formula only applies to right-angled triangles. For oblique triangles, use the law of cosines instead.

Using trigonometric functions in a right triangle calculation

Trigonometric functions connect an acute angle to the ratio of two sides:

• sin(angle) = opposite / hypotenuse
• cos(angle) = adjacent / hypotenuse
• tan(angle) = opposite / adjacent

If you know one side and one acute angle, pick the trig function that includes the known side and the side you want. Rearrange and solve.

For example, if you know the hypotenuse and want the opposite side, multiply: opposite = hypotenuse × sin(angle).

Special Right Triangle Angles: 45-45-90 and 30-60-90

Two triangle types have fixed side ratios that make mental math possible.

45-45-90 triangle:

• Both legs have equal lengths.
• Hypotenuse = leg × √2.
• Example: legs of 5 each produce a hypotenuse of 5√2 ≈ 7.07.

30-60-90 triangle:

• The side opposite 30° is the shortest.
• The side opposite 60° = short side × √3.
• Hypotenuse = short side × 2.
• Example: short side of 4 gives sides of 4, 4√3 ≈ 6.93, and 8.

These ratios appear constantly in geometry, construction, and standardized tests. The calculator recognizes these patterns automatically when you input the corresponding angles.

Triangle Angle Calculation: How Angles Add to 180°

The interior angles of any triangle always sum to exactly 180°. This is the most fundamental rule in triangle geometry.

If you know two angles, the third is simply 180° minus the other two. The calculator uses this rule as a first step in almost every solution path.

Internal and external angles

An interior angle is the angle inside the triangle at each vertex. An exterior angle is formed by one side of the triangle and the extension of an adjacent side.

Key facts:

• Each exterior angle equals 180° minus its adjacent interior angle.
• Each exterior angle also equals the sum of the two non-adjacent interior angles.
• The three exterior angles of any triangle sum to 360°.

These relationships help verify your results. If the angles the calculator returns do not follow these rules, the input values may describe an impossible triangle.

Side Length Calculation Using the Law of Sines and Cosines

When a triangle is not right-angled, you need the law of sines or the law of cosines (or both) to solve it.

Law of sines:

a / sin(A) = b / sin(B) = c / sin(C)

This ratio is constant for all three side-angle pairs. Use it when you know one side and its opposite angle plus at least one other value.

Law of cosines:

c² = a² + b² − 2ab × cos(C)

This generalizes the Pythagorean theorem. When angle C is 90°, the cos(C) term drops to zero and you get a² + b² = c². Use it when you know two sides and the angle between them, or all three sides.

The calculator picks the correct law based on your input. You do not need to decide which formula applies.

Triangle Area, Perimeter, and Height

Perimeter is the simplest measurement: add all three side lengths.

Area can be calculated several ways depending on known values:

• Base and height: Area = ½ × base × height
• Two sides and the included angle: Area = ½ × a × b × sin(C)
• Three sides (Heron's formula): First find s = (a + b + c) / 2, then Area = √(s(s−a)(s−b)(s−c))

Height (altitude) is the perpendicular distance from a vertex to the opposite side. Each triangle has three altitudes, one from each vertex. The calculator returns the altitude relative to each side.

A gardener planting tulip bulbs around a triangular flower bed with sides of 5 m, 6 m, and 10 m would first find the perimeter: 5 + 6 + 10 = 21 m. At 8 bulbs per meter, that is 21 × 8 = 168 bulbs.

Area of a triangle from two sides and one angle

The formula Area = ½ × a × b × sin(C) is especially handy when you do not know the height. You only need the length of two sides and the angle between them.

This works because b × sin(C) equals the height relative to side a. The sine function converts the angled side into a perpendicular distance automatically.

Incircle, circumcircle, and centroid

These are three important geometric features of a triangle:

• Incircle. The largest circle that fits inside the triangle, touching all three sides. Its radius (inradius) = Area / s, where s is the semi-perimeter.
• Circumcircle. The circle passing through all three vertices. Its radius (circumradius) = (a × b × c) / (4 × Area).
• Centroid. The point where the three medians intersect. A median connects a vertex to the midpoint of the opposite side. The centroid is always located one-third of the way from each side toward the opposite vertex.

These values help in engineering, design, and construction planning.

Triangle Types: Equilateral, Isosceles, Acute, and Obtuse Triangles

Triangles are classified by their sides and their angles.

By sides:

• Equilateral. All three sides have equal lengths. All angles are 60°.
• Isosceles. Two sides are equal. The two angles opposite those sides are also equal.
• Scalene. No sides are equal. All three angles differ.

By angles:

• Acute. All interior angles are less than 90°.
• Right. One angle is exactly 90°.
• Obtuse. One angle is greater than 90°.

The longest side of the triangle is always opposite the largest angle. In a right triangle, that longest side is called the hypotenuse. In an obtuse triangle, the longest edge sits opposite the obtuse angle.

A triangle cannot have equal sides and also be right-angled. An equilateral triangle requires all 60° angles, and an isosceles right triangle has angles of 45°, 45°, and 90° with only two sides equal.

Right Triangle Calculator vs. General Triangle Calculator

The right triangle calculator and the general triangle calculator solve different scopes of problems.

If you know you are working with a right-angled triangle, the right triangle calculator is faster because it applies simpler formulas. If the triangle might not contain a 90° angle, use the general calculator instead.

This page handles both. Enter a 90° angle and it switches to right triangle mode automatically.

Triangle Calculation Symbol and Formula Reference

Vertex, angle symbol, and side length labels

Standard triangle labeling uses uppercase letters for vertices and angles, and lowercase letters for the opposite sides.

• Vertices: A, B, C (the three corner points)
• Angles: A (or α, alpha) at vertex A, B (or β, beta) at vertex B, C at vertex C
• Sides: a is opposite angle A, b is opposite angle B, c is opposite angle C

In a right triangle, the right angle is typically placed at vertex C, making c the hypotenuse.

Three vertices define three line segments (the sides). Angle bisectors, medians, and altitudes are the main interior line segments used in advanced calculations.

Radian vs. degree angle input

Angles can be measured in degrees or radians.

• Degrees: A full circle is 360°. A right angle is 90°.
• Radians: A full circle is 2π (approximately 6.283). A right angle is π/2 (approximately 1.571).

To convert: radians = degrees × (π / 180). To go the other way: degrees = radians × (180 / π).

The calculator accepts both. Make sure your input matches the selected unit. Entering 1.57 when you mean 90° will produce very different results.

Calculation Limitations: What This Triangle Calculator Estimates

This calculator gives mathematically precise answers based on the values you enter. However, keep a few limitations in mind:

• Input accuracy matters. If your measured side lengths or angles are approximate, the results will inherit that imprecision. Small measurement errors can cause noticeable differences in the output, especially for obtuse or very narrow triangles.
• Ambiguous case (SSA). Two sides and a non-included angle can produce zero, one, or two valid triangles. The calculator shows all valid solutions, but you must determine which one applies to your situation.
• Not a substitute for professional work. For structural engineering, surveying, or any application where safety depends on accuracy, consult a qualified professional. This tool is a planning and learning aid.
• Floating point rounding. The calculator may round results to a set number of decimal places. For most practical purposes this is negligible, but do not treat the output as exact to infinite precision.

If you need to solve a triangle for homework, DIY projects, or quick verification, this calculator handles it reliably. For high-stakes applications, use the results as a starting estimate and verify with professional tools.