Free Triangle Calculator - Calculate Sides, Angles, Area & Perimeter

Calculate any triangle's missing sides, angles, area, and perimeter instantly. Supports SSS, SAS, ASA/AAS modes with step-by-step solutions.

What Is a Triangle Calculator?

A triangle calculator is a mathematical tool that helps you solve triangles by computing unknown sides, angles, area, and perimeter. Whether you know three sides (SSS), two sides and an angle (SAS), or two angles and a side (ASA/AAS), this calculator uses fundamental trigonometric laws to determine all remaining properties of the triangle.

Our triangle calculator applies the Law of Cosines, Law of Sines, and Heron's formula to deliver precise results with detailed step-by-step explanations. It also classifies the triangle by its sides (equilateral, isosceles, scalene) and by its angles (acute, right, obtuse).

Types of Triangles

Triangles can be classified in two ways: by their sides and by their angles.

Classification by Sides

Classification by Angles

How to Use the Triangle Calculator

SSS Mode (Three Sides)

1. Enter the lengths of all three sides (a, b, c)
2. The calculator validates the triangle inequality theorem
3. Results include all three angles, area, and perimeter
4. The solution uses the Law of Cosines to find each angle

Example: For sides a = 5, b = 7, c = 9:

• cos(A) = (b² + c² - a²) / (2bc) = (49 + 81 - 25) / 126 = 0.8333
• A = arccos(0.8333) ≈ 33.56°
• The calculator computes all remaining angles similarly

SAS Mode (Two Sides + Included Angle)

1. Enter two sides (a, b) and the angle between them (C)
2. The calculator uses the Law of Cosines to find the third side
3. Then the Law of Sines or Cosines finds the remaining angles
4. Area is calculated using: Area = ½ × a × b × sin(C)

Example: Side a = 8, side b = 6, angle C = 45°:

• c² = a² + b² - 2ab·cos(C) = 64 + 36 - 96 × cos(45°)
• c = √(100 - 67.88) = √32.12 ≈ 5.67

ASA/AAS Mode (Two Angles + One Side)

1. Enter two angles (A, B) and one side (a)
2. The third angle is found: C = 180° - A - B
3. The Law of Sines finds the remaining sides: b = a × sin(B) / sin(A)
4. Area is computed using Heron's formula

Example: Angle A = 40°, angle B = 60°, side a = 10:

• C = 180° - 40° - 60° = 80°
• b = 10 × sin(60°) / sin(40°) ≈ 13.47
• c = 10 × sin(80°) / sin(40°) ≈ 15.32

Area Calculator Mode

Choose between two methods:

Base and Height:

• Area = ½ × base × height
• Simplest method when base and height are known

Heron's Formula:

• Enter all three sides
• Semi-perimeter: s = (a + b + c) / 2
• Area = √(s(s-a)(s-b)(s-c))
• Works for any triangle when only sides are known

Triangle Formulas

Law of Cosines

The Law of Cosines relates the lengths of a triangle's sides to the cosine of one of its angles:

c² = a² + b² - 2ab·cos(C)

This formula generalizes the Pythagorean theorem and works for any triangle, not just right triangles. It is especially useful when you know all three sides (SSS) or two sides and the included angle (SAS).

Rearranged to find an angle:

cos(C) = (a² + b² - c²) / (2ab)

Law of Sines

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

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

This is most useful when you know two angles and a side (ASA or AAS), or two sides and an angle opposite one of them (SSA — though this case can be ambiguous).

Heron's Formula

Heron's formula calculates the area of a triangle when only the three side lengths are known:

s = (a + b + c) / 2 (semi-perimeter)

Area = √(s × (s-a) × (s-b) × (s-c))

This elegant formula, attributed to Hero of Alexandria, works for any type of triangle without needing to know any angles or heights.

Area = ½ × Base × Height

The most fundamental triangle area formula:

Area = ½ × b × h

Where b is the base length and h is the perpendicular height from the base to the opposite vertex.

Perimeter

Perimeter = a + b + c

The sum of all three side lengths.

Common Triangle Problems with Examples

Problem 1: Finding a Missing Side

Given a triangle with b = 12, c = 15, and angle A = 70°, find side a.

Using the Law of Cosines: a² = b² + c² - 2bc·cos(A) a² = 144 + 225 - 2(12)(15)cos(70°) a² = 369 - 360 × 0.3420 a² = 369 - 123.12 = 245.88 a ≈ 15.68

Problem 2: Finding a Missing Angle

Given a triangle with a = 7, b = 10, c = 13, find angle C.

Using the Law of Cosines: cos(C) = (a² + b² - c²) / (2ab) cos(C) = (49 + 100 - 169) / (140) cos(C) = -20/140 = -0.1429 C = arccos(-0.1429) ≈ 98.21°

Since C > 90°, this is an obtuse triangle.

Problem 3: Calculating Area with Three Sides

Given a triangle with a = 5, b = 6, c = 7, find the area.

s = (5 + 6 + 7) / 2 = 9 Area = √(9 × 4 × 3 × 2) = √216 ≈ 14.70

Problem 4: Right Triangle Check

Given sides 3, 4, 5 — is this a right triangle? 3² + 4² = 9 + 16 = 25 5² = 25 25 = 25 ✓ — Yes, it is a right triangle!

The Triangle Inequality Theorem

The triangle inequality theorem states that for any triangle with sides a, b, and c:

• a + b > c
• a + c > b
• b + c > a

In other words, the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. This is a fundamental test to determine whether three given lengths can form a valid triangle.

Example: Can sides 3, 4, 8 form a triangle?

• 3 + 4 = 7, which is not greater than 8
• Therefore, 3, 4, 8 cannot form a triangle

Our calculator automatically checks the triangle inequality before performing any calculations.

Real-World Applications of Triangle Calculations

Architecture and Construction

Triangles are the strongest geometric shape because they distribute weight evenly. Architects use triangle calculations to design roof trusses, bridge supports, and structural frameworks. The rigidity of triangles makes them essential in ensuring buildings can withstand loads and forces from multiple directions.

Engineering

Civil and mechanical engineers rely on triangle calculations for designing everything from aircraft wings to automobile frames. Surveying, a critical engineering discipline, uses triangulation — measuring angles and distances in triangles — to map terrain and establish property boundaries with high precision.

Navigation and GPS

Triangulation is the backbone of GPS technology. By measuring the distance from a receiver to at least three satellites, GPS systems form virtual triangles and calculate your exact position on Earth. The Law of Cosines plays a central role in converting satellite signal timing data into accurate geographic coordinates.

Surveying and Land Measurement

Land surveyors use triangulation networks to measure large areas with extraordinary accuracy. Starting from a known baseline, surveyors measure angles to distant points and use trigonometric calculations to determine distances and positions. This method has been used for centuries to map countries and continents.

Art, Design, and Computer Graphics

In computer graphics, all 3D models are composed of triangles (a process called tessellation). Game engines and rendering software use triangle calculations for lighting, collision detection, and physics simulations. Artists and designers also use triangular compositions to create visually balanced and dynamic layouts.

Physics and Science

Triangles appear throughout physics, from force vector diagrams (where forces are resolved into triangular components) to optics (where light refraction angles are calculated). Understanding triangle properties is essential for solving problems in mechanics, electromagnetism, and wave theory.

Frequently Asked Questions

How do I find the area of a triangle?

There are several ways to calculate a triangle's area. The simplest is Area = ½ × base × height, where the height is the perpendicular distance from the base to the opposite vertex. If you only know the three sides, use Heron's formula: Area = √(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2 is the semi-perimeter. If you know two sides and the included angle, use Area = ½ × a × b × sin(C).

How do I use the Law of Cosines?

The Law of Cosines is c² = a² + b² - 2ab·cos(C). Use it when you know two sides and the included angle (SAS) to find the third side, or when you know all three sides (SSS) and want to find an angle. To find an angle, rearrange to cos(C) = (a² + b² - c²) / (2ab) and take the inverse cosine.

What is Heron's formula?

Heron's formula lets you calculate the area of any triangle when you know only the three side lengths. First compute the semi-perimeter s = (a + b + c) / 2, then calculate Area = √(s × (s-a) × (s-b) × (s-c)). It works for all triangle types — equilateral, isosceles, scalene, right, acute, or obtuse.

How do I check if three sides can form a triangle?

Use the triangle inequality theorem: the sum of any two sides must be strictly greater than the third side. Check all three conditions: a + b > c, a + c > b, and b + c > a. If all three are true, the sides can form a valid triangle. If even one condition fails, no triangle is possible.

What is the triangle inequality theorem?

The triangle inequality theorem states that in any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This is because the shortest distance between two points is a straight line — the third side must be shorter than the "detour" of going through the other vertex via the other two sides.

How do I find missing angles in a triangle?

If you know two angles, the third is simply 180° - A - B (since all angles in a triangle sum to 180°). If you know all three sides, use the Law of Cosines: cos(A) = (b² + c² - a²) / (2bc), then A = arccos(result). If you know two sides and a non-included angle, use the Law of Sines: sin(B) = b × sin(A) / a.

Can a triangle have two right angles?

No. Since the sum of all three angles in a triangle is exactly 180°, having two right angles (each 90°) would leave 0° for the third angle, which is impossible. A triangle can have at most one right angle or one obtuse angle.

What is the difference between ASA and AAS?

ASA (Angle-Side-Angle) means you know two angles and the side between them. AAS (Angle-Angle-Side) means you know two angles and a side that is not between them. Both can be solved the same way: first find the third angle (C = 180° - A - B), then use the Law of Sines to find the remaining sides.

Why Use Our Triangle Calculator

Our triangle calculator is designed for students, teachers, engineers, architects, and anyone who needs fast, accurate triangle calculations. Here is what makes it stand out:

• Multiple calculation modes — SSS, SAS, ASA/AAS, and dedicated area calculation
• Step-by-step solutions — Every calculation shows the full working process so you can learn and verify
• Triangle classification — Automatically identifies equilateral, isosceles, scalene, acute, right, or obtuse triangles
• Visual diagram — See your triangle rendered with labeled sides and angles
• Instant validation — The triangle inequality theorem is checked automatically before computing results
• Heron's formula and more — Multiple area calculation methods to suit the data you have available
• Free and no sign-up — Use it as many times as you need with no account required
• Mobile-friendly — Works perfectly on phones, tablets, and desktops

Whether you are solving homework problems, preparing for exams, or working on a professional project, our triangle calculator gives you the answers you need with the confidence of seeing every step.