Triangle Calculator

Triangle Basics

A triangle is a polygon with three vertices and three sides. The vertices are typically denoted as A, B, and C, and the opposite sides as a, b, and c respectively. For right triangles specifically, try our right triangle calculator.

The interior angles of any triangle always sum to 180 degrees (or π radians). This is known as the angle sum property and is one of the most fundamental theorems in Euclidean geometry. It holds true for every triangle regardless of its shape or size, from tiny microscopic triangles to vast triangles spanning astronomical distances.

Triangles are classified either by their side lengths (equilateral, isosceles, scalene) or by their angles (acute, right, obtuse). A triangle cannot have more than one right angle or more than one obtuse angle, because the sum of angles would exceed 180°. The smallest angle is always opposite the shortest side, and the largest angle is always opposite the longest side, a relationship that holds true for all triangles and is a direct consequence of the Law of Sines.

Types of Triangles

Equilateral Triangle: All three sides and angles are equal (each angle is 60°).

Isosceles Triangle: Two sides and two angles are equal.

Scalene Triangle: All sides and angles are different.

Right Triangle: One angle is exactly 90°. The longest side is called the hypotenuse. Use our right triangle calculator for specialized right triangle calculations.

Obtuse Triangle: One angle is greater than 90°.

Acute Triangle: All angles are less than 90°.

Pythagorean Theorem

The Pythagorean theorem applies only to right triangles. For a right triangle with legs a and b, and hypotenuse c. You can verify your calculations with our Pythagorean theorem calculator:

Example: A 3-4-5 right triangle has sides 3, 4, and 5 units.

Law of Sines

The Law of Sines relates the sides of a triangle to the sines of their opposite angles:

This is useful when you know two angles and one side (AAS or ASA), or two sides and an angle opposite one of them (SSA). The constant ratio a/sin(A) is actually equal to 2R, where R is the circumradius of the triangle. This provides a direct link between the side lengths and the circumscribed circle, meaning the Law of Sines can also be used to find the circumradius of any triangle when at least one side and its opposite angle are known.

The Law of Sines is particularly powerful because it works for any triangle, not just right triangles. It is derived from the fact that the area of a triangle can be expressed in three different ways using the sine of each angle: Area = ½bc·sin(A) = ½ac·sin(B) = ½ab·sin(C). Setting these equal to each other and dividing through by the common factor yields the Law of Sines relationship.

Law of Cosines

The Law of Cosines relates the three sides of a triangle to one of its angles:

Solving for the angle:

This is useful when you know all three sides (SSS) or two sides and the included angle (SAS). The Law of Cosines is a generalization of the Pythagorean theorem. In fact, when angle A is 90°, cos(90°) = 0, so the formula reduces to a² = b² + c², which is the Pythagorean theorem. This shows that the Pythagorean theorem is a special case of the Law of Cosines that applies only to right triangles, while the Law of Cosines works for any triangle regardless of angle type.

Triangle Area

You can also use our area calculator for other shape areas.

Base and Height: Area = ½ × base × height

Heron's Formula: When all three sides are known, use Heron's formula:

Two Sides and Included Angle:

How to Use the Triangle Calculator

Using the triangle calculator is straightforward. Enter any three values of your triangle, ensuring at least one is a side length, and click Calculate. The calculator will determine all remaining properties automatically using the Law of Sines, Law of Cosines, and other geometric formulas.

You can enter angles in either degrees or radians by selecting your preferred unit from the Angle Unit dropdown. For common angles in radian mode, try values like pi/2 (90°), pi/3 (60°), or pi/4 (45°). The visual triangle diagram updates in real time to show your triangle's shape. For unit conversions, use our unit conversion calculator.

For more specialized calculations, you can also use our right triangle calculator for right-angle problems or our Pythagorean theorem calculator for finding missing sides of right triangles. Our slope calculator is useful for calculating line slopes and angles of elevation.

Common Triangle Problems and Solutions

Finding the missing side of a right triangle: If you know two sides of a right triangle, use the Pythagorean theorem: a² + b² = c². For example, if side a = 3 and side b = 4, then c = √(3² + 4²) = √25 = 5.

Finding the area with limited information: If you only know two sides and the included angle, use the formula Area = ½ab·sin(C). For a triangle with sides 5 and 7 units and an included angle of 30°, the area is ½ × 5 × 7 × sin(30°) = 8.75 square units.

Classifying a triangle by its sides: Given three side lengths, check if the triangle is equilateral (all equal), isosceles (two equal), or scalene (all different). Then use the Law of Cosines to find the largest angle and determine if the triangle is acute, right, or obtuse.

Solving SSS triangles: When all three sides are known but no angles are given, use the Law of Cosines to find each angle. For example, with sides a=7, b=8, c=9, you can find angle A using cos(A) = (b² + c² - a²) / 2bc. Use our Pythagorean theorem calculator for right triangle problems, our area calculator for checking your area calculations, and our volume calculator for three-dimensional geometry problems involving triangles.

Special Right Triangles

30-60-90 Triangle: In a right triangle with angles 30°, 60°, and 90°, the sides follow a fixed ratio. If the shortest leg (opposite 30°) is x, the hypotenuse is 2x, and the longer leg (opposite 60°) is x√3. This ratio makes 30-60-90 triangles common in geometry problems and real-world applications such as construction and design.

45-45-90 Triangle: An isosceles right triangle with angles 45°, 45°, and 90°. If each leg is x, the hypotenuse is x√2. Both legs are equal because the base angles are equal. These triangles appear frequently in architecture, engineering drawings, and trigonometric calculations.

3-4-5 Triangle: A right triangle with sides 3, 4, and 5 units. Since 3² + 4² = 9 + 16 = 25 = 5², it satisfies the Pythagorean theorem perfectly. The 3-4-5 triangle is widely used in construction for checking right angles and squaring corners. Any multiple (6-8-10, 9-12-15, etc.) also forms a right triangle.

Our right triangle calculator can quickly solve any special right triangle by entering just two known values.

Triangle Congruence and Similarity

Triangle Congruence: Two triangles are congruent if they have the same shape and size, meaning all corresponding sides and angles are equal. There are five congruence criteria: SSS (all three sides equal), SAS (two sides and the included angle equal), ASA (two angles and the included side equal), AAS (two angles and a non-included side equal), and HL (hypotenuse-leg for right triangles). Congruent triangles can be mapped onto each other through rotation, reflection, or translation.

Triangle Similarity: Two triangles are similar if they have the same shape but not necessarily the same size. Corresponding angles are equal, and corresponding sides are in proportion. Similarity criteria include AA (two angles equal), SAS (two sides in proportion and the included angle equal), and SSS (all three sides in proportion). If triangle ABC is similar to triangle DEF, then AB/DE = BC/EF = AC/DF. Similar triangles are used extensively in indirect measurement, such as determining the height of a building using shadows.

Understanding congruence and similarity is essential for advanced geometry and trigonometry. Use our triangle calculator to verify your congruence and similarity proofs by checking side lengths and angles.

Real-World Applications of Triangles

Construction and Architecture: Triangles are the most structurally stable shape. Roof trusses, bridges, cranes, and scaffolding all use triangular frameworks because triangles distribute weight evenly and resist deformation. The triangle calculator helps architects and engineers quickly verify angles and side lengths for structural designs.

Navigation and Surveying: Surveyors use triangulation to measure distances and map terrain. By measuring a baseline distance and two angles, they can calculate the distance to any point using the Law of Sines. GPS systems use trilateration, which involves solving triangles between satellites and a receiver to determine precise location coordinates on Earth.

Astronomy: Astronomers use parallax, a triangulation method, to measure distances to nearby stars. By observing a star from two different positions in Earth's orbit around the Sun, they create a triangle and calculate the star's distance using trigonometry. The tiny angular shift, measured in arcseconds, reveals vast cosmic distances.

Computer Graphics: 3D models are built from thousands of triangles joined together in a mesh. Each triangle's vertices define a surface that can be textured, lit, and rendered. Video games, CGI movies, and CAD software all rely on triangle-based rendering for realistic 3D graphics. Programmers use triangle geometry to calculate lighting angles, surface normals, and collision detection in virtual environments.

Our slope calculator and area calculator complement these real-world applications by providing additional geometric calculations.

Trigonometric Ratios in Triangles

Trigonometric ratios relate the angles of a right triangle to the ratios of its sides. The three primary ratios are sine (sin), cosine (cos), and tangent (tan):

The mnemonic SOH CAH TOA helps remember these relationships. For a right triangle with angle θ, the opposite side is across from θ, the adjacent side is next to θ (but not the hypotenuse), and the hypotenuse is the longest side opposite the right angle.

Inverse Trigonometric Functions: To find an angle when you know two sides, use inverse trigonometric functions: arcsin (sin⁻¹), arccos (cos⁻¹), and arctan (tan⁻¹). For example, if the opposite side is 3 and the hypotenuse is 5, then sin(θ) = 3/5 = 0.6, so θ = arcsin(0.6) ≈ 36.87°. The triangle calculator uses these functions internally to compute angles from side lengths. These inverse functions are essential in navigation for calculating bearing angles, in physics for determining launch angles of projectiles, and in engineering for computing slopes and gradients of structures.

Applications: Trigonometric ratios are used in physics to resolve forces into components, in engineering to calculate slopes and gradients, and in navigation to determine bearings and distances. Our triangle calculator handles all trigonometric calculations automatically, so you can focus on applying the results to your specific problem.

Common Mistakes When Solving Triangles

Using the wrong formula: The most common mistake is applying the Pythagorean theorem to non-right triangles. The Pythagorean theorem only works for right triangles. For other triangles, use the Law of Cosines or Law of Sines depending on what values you know. Our triangle calculator automatically selects the correct formula based on your inputs.

Forgetting to check the triangle inequality: Not every set of three values can form a triangle. The sum of any two sides must be greater than the third side. If you enter values like sides 1, 2, and 5, no triangle exists because 1 + 2 is not greater than 5. The calculator checks this condition and alerts you if the inputs are invalid.

Ambiguous case (SSA): When given two sides and a non-included angle (SSA), there can be zero, one, or two possible triangles. This is known as the ambiguous case. If the angle is acute and the opposite side is shorter than the adjacent side but longer than the height, there are two possible triangles. The calculator handles this case and displays all valid solutions.

Angle unit confusion: Mixing degrees and radians is a common error. Always verify which unit your calculator or problem uses. The triangle calculator lets you choose between degrees and radians, and displays all results in the selected unit. Remember that 180° = π radians, so to convert from degrees to radians, multiply by π/180.

Rounding too early: When solving multi-step triangle problems, rounding intermediate values can lead to significant errors in the final result. Always keep full precision during intermediate calculations and round only the final answer. The triangle calculator maintains high precision throughout all calculations to ensure accurate results.

Tips for Using the Triangle Calculator Effectively

Enter precise values: For the most accurate results, enter your known values with as much precision as possible. The calculator supports decimal values and common fractions. In radian mode, you can enter expressions like pi/2, pi/3, pi/4, or pi/6 for common angles.

Check the visual diagram: The interactive triangle diagram updates to reflect your entered values, showing a scaled representation of the triangle. Use this visual feedback to verify that your inputs produce a triangle matching your expectations. If the diagram looks unexpected, double-check your entered values for errors.

Review the calculation steps: Click the Show Calculation Steps button to see the detailed step-by-step solution. This is invaluable for learning how triangle calculations work and for verifying your manual work. Each step shows which formula was used and the intermediate results.

Use the correct input combinations: Remember that you need at least one side to solve a triangle. Three angles alone are not enough because they only determine the triangles shape but not its size (similar triangles). Make sure at least one of your three inputs is a side length for the calculator to produce a unique solution.

Cross-check with related calculators: Use our right triangle calculator for specialized right triangle problems and our Pythagorean theorem calculator for quick right-triangle side calculations. The area calculator can help verify your area results using different methods.

Centroid, Incenter, Circumcenter, and Orthocenter

Every triangle has several special points called centers, each with unique properties. Understanding these centers is important for advanced geometry and engineering applications.

Centroid (G): The intersection point of the three medians. It is the center of mass of the triangle. If you cut a triangle out of cardboard, it balances perfectly at its centroid. The centroid divides each median in a 2:1 ratio, with the longer segment being from the vertex to the centroid. The coordinates of the centroid are the average of the three vertices coordinates.

Incenter (I): The intersection point of the three angle bisectors. It is the center of the incircle, the largest circle that fits inside the triangle touching all three sides. The incenter is equidistant from all three sides, and that distance is the inradius (r) which our calculator computes. The incenter always lies inside the triangle regardless of the triangles type.

Circumcenter (O): The intersection point of the three perpendicular bisectors. It is the center of the circumcircle that passes through all three vertices. The distance from the circumcenter to any vertex is the circumradius (R). In an acute triangle, the circumcenter lies inside; in a right triangle, it lies at the midpoint of the hypotenuse; in an obtuse triangle, it lies outside.

Orthocenter (H): The intersection point of the three altitudes. In an acute triangle, the orthocenter lies inside. In a right triangle, it is at the vertex of the right angle. In an obtuse triangle, it lies outside. The orthocenter has many interesting properties, including the fact that it forms an Euler line with the centroid and circumcenter in any non-equilateral triangle.

Triangle Perimeter and Semi-perimeter

Perimeter: The perimeter of a triangle is the total distance around its three sides. It is calculated by simply adding the lengths of all three sides: P = a + b + c. The perimeter is measured in linear units (meters, feet, inches, etc.) and represents the boundary length of the triangle. In real-world applications, knowing the perimeter is essential for calculating the amount of fencing needed for a triangular plot, the length of trim for a triangular window, or the total border length for a triangular garden bed.

Semi-perimeter: The semi-perimeter is half of the perimeter: s = (a + b + c) / 2. While seemingly a simple derivation, the semi-perimeter is a critical value appearing in many triangle formulas. Most notably, it is used in Heron's formula for calculating the area of a triangle when all three sides are known. The semi-perimeter also appears in formulas for the inradius (r = Area / s) and the distance from the incenter to each side.

For example, consider a triangle with sides 7, 8, and 9 units. The perimeter is 7 + 8 + 9 = 24 units. The semi-perimeter is 24 / 2 = 12 units. Using Heron's formula: Area = √(12 × (12-7) × (12-8) × (12-9)) = √(12 × 5 × 4 × 3) = √720 ≈ 26.83 square units. The triangle calculator computes all these values instantly and displays them in the results section along with the inradius and circumradius.

Understanding the Triangle Inequality Theorem

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This must hold true for all three combinations: a + b > c, a + c > b, and b + c > a. If any of these conditions fails, the three lengths cannot form a triangle.

Why it matters: The triangle inequality is not just a theoretical concept; it has practical implications in many fields. In geometry, it ensures that three line segments can actually form a closed shape. In navigation, it limits the possible paths between three points. In computer science, the triangle inequality is used to optimize algorithms for nearest-neighbor search and network routing.

Examples: The lengths 3, 4, and 5 satisfy the triangle inequality because 3 + 4 > 5, 3 + 5 > 4, and 4 + 5 > 3, so they form a valid right triangle. The lengths 1, 2, and 5 do NOT satisfy the inequality because 1 + 2 is not greater than 5, so no triangle exists. The special case where a + b = c is called a degenerate triangle, where all three points lie on a straight line with zero area.

Relationship to angles: The triangle inequality is closely related to the angles of a triangle. The largest side is always opposite the largest angle, and the smallest side is always opposite the smallest angle. If one side is too long relative to the other two, the opposite angle would need to be greater than 180° to form a closed shape, which is impossible in Euclidean geometry. Our triangle calculator automatically validates your inputs against the triangle inequality theorem and alerts you if the values cannot form a valid triangle.

Final Thoughts on Triangle Calculations

Triangles are one of the most fundamental shapes in geometry and appear in virtually every field of science, engineering, and mathematics. Understanding how to calculate triangle properties is an essential skill that extends far beyond the classroom. From the simplest Pythagorean theorem to advanced applications in physics, computer graphics, and astronomy, triangle calculations form the backbone of countless technologies and discoveries that shape our modern world.

Our triangle calculator is designed to handle all common triangle-solving scenarios, including SSS, SAS, SSA, ASA, and AAS configurations. It automatically selects the appropriate formulas, checks for the triangle inequality, handles the ambiguous SSA case, and provides detailed step-by-step solutions. Whether you are a student learning geometry, a professional engineer designing a structure, a surveyor mapping terrain, or a hobbyist working on a DIY project, this calculator provides the accuracy and reliability you need for your triangle calculations.

We encourage you to explore the calculation steps feature to deepen your understanding of triangle geometry. By seeing exactly how each value is derived, you can learn the underlying principles and apply them confidently in your own work. The interactive visual diagram also helps build intuition about how changes in side lengths and angles affect the overall shape of a triangle. Practice with different input combinations, experiment with special triangles like the 30-60-90 and 45-45-90, and use the results to verify your manual calculations.

Remember that consistent practice is the key to mastering triangle geometry. The more you work with triangles, the more intuitive the relationships between sides, angles, and other properties become. Use this calculator as a learning tool to check your work and build confidence in your geometric reasoning skills.