Triangle Calculator — Free Online Triangle Solver

How to Use the Triangle Calculator

1. Select the input method: Choose from five solving methods based on the information you have. SSS requires three side lengths. SAS requires two sides and the angle between them. ASA requires two angles and the side between them. AAS requires two angles and a side not between them. SSA requires two sides and an angle not between them.
2. Enter known values: Type your measurements into the three input fields. The labels automatically update to match the selected method. Side lengths must be positive numbers. Angles must be entered in degrees and must be between 0 and 180 degrees. The input fields accept decimal values for precise calculations.
3. Review the results: The results panel instantly shows all computed values including all three sides, all three angles, the total area (computed via Heron's formula), the perimeter, and the three altitudes (heights to each side). No button press is needed as results update in real time.
4. Validate your triangle: If your input values cannot form a valid triangle (for example, the triangle inequality is violated or the angles sum exceeds 180 degrees), the calculator will display a message indicating that the values are invalid. Adjust your inputs and try again.

When switching between methods, the calculator loads sensible default values so you can immediately see how each method works. Experiment freely with different values and methods to develop your understanding of triangle geometry.

Triangle Formulas

Variables Explained

• a, b, c: The three side lengths of the triangle. Side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. All side lengths must be positive.
• A, B, C: The three interior angles measured in degrees. Each angle must be greater than 0 and less than 180 degrees. Their sum always equals exactly 180 degrees.
• s: The semi-perimeter, equal to half the perimeter. Used in Heron's formula for calculating area from side lengths alone.
• h_a, h_b, h_c: The three altitudes (heights) of the triangle. Each height is measured perpendicular to its corresponding base.

Step-by-Step Example (SSS Method)

Solve a triangle with sides a = 5, b = 7, c = 8:

1. Calculate the semi-perimeter: s = (5 + 7 + 8) / 2 = 10
2. Apply Heron's formula: Area = sqrt(10 x 5 x 3 x 2) = sqrt(300) = 17.32 sq units
3. Find angle A using the Law of Cosines: cos(A) = (7² + 8² - 5²) / (2 x 7 x 8) = 0.7857, so A = 38.21°
4. Find angle B: cos(B) = (5² + 8² - 7²) / (2 x 5 x 8) = 0.5, so B = 60°
5. Find angle C: C = 180 - 38.21 - 60 = 81.79°
6. Calculate heights: h_a = 2 x 17.32 / 5 = 6.93, h_b = 2 x 17.32 / 7 = 4.95, h_c = 2 x 17.32 / 8 = 4.33

Practical Examples

Example 1: Nathan's Backyard Triangular Garden

Nathan is building a triangular garden bed in his backyard. He measured the three sides as 12 feet, 15 feet, and 18 feet. He needs to know the area to calculate how much soil to buy and the angles to cut the border pieces correctly.

• Semi-perimeter: s = (12 + 15 + 18) / 2 = 22.5
• Area = sqrt(22.5 x 10.5 x 7.5 x 4.5) = sqrt(7,971.56) = 89.28 sq ft
• Angles: A = 41.41°, B = 56.25°, C = 82.34°

Nathan needs approximately 89.28 square feet of soil coverage. Since garden soil is typically sold by the cubic foot, he would multiply this area by his desired depth (for example, 89.28 x 0.5 feet = 44.64 cubic feet for 6 inches of soil). The angle measurements help him cut the wooden border pieces at the correct angles.

Example 2: Sarah's Surveying Problem

Sarah is a land surveyor measuring a triangular plot. She can measure two sides as 45 meters and 62 meters, and the included angle between them as 73 degrees (SAS method). She needs the full triangle dimensions for the property boundary report.

• Third side: c = sqrt(45² + 62² - 2 x 45 x 62 x cos(73°)) = sqrt(2025 + 3844 - 1631.4) = 65.10 m
• Perimeter: 45 + 62 + 65.10 = 172.10 m
• Area = (1/2) x 45 x 62 x sin(73°) = 1,334.26 sq m

Sarah can now report the full property boundary: three sides of 45 m, 62 m, and 65.10 m with a total area of 1,334.26 square meters (approximately 0.33 acres). The perimeter of 172.10 meters tells her how much fencing the property would need. For calculating the area of rectangular plots, try our area calculator.

Example 3: Mark's Roof Truss Design

Mark is designing a roof truss. He knows the two base angles are 35 degrees and 55 degrees, and the base (horizontal span) is 24 feet. Using the ASA method, he calculates the rafter lengths and ridge height.

• Third angle: C = 180 - 35 - 55 = 90° (this is a right triangle)
• Side a (opposite 35°): a = 24 x sin(35°) / sin(90°) = 13.77 ft
• Side b (opposite 55°): b = 24 x sin(55°) / sin(90°) = 19.66 ft
• Area: (1/2) x 13.77 x 19.66 = 135.36 sq ft

Mark now knows both rafter lengths (13.77 feet and 19.66 feet) and can cut them accordingly. The 90-degree angle at the ridge confirms this is a right triangle, simplifying his construction. The area of 135.36 square feet helps him estimate roofing materials for each triangular face of the roof.

Example 4: Diana's Navigation Calculation

Diana is hiking and uses triangulation to estimate the distance to a mountain peak. She takes bearings from two points 500 meters apart. The angles from each point to the peak are 72 degrees and 63 degrees (AAS method with A = 72°, B = 63°, a = side opposite 72° is unknown, and the known side c = 500 m is opposite C).

• Third angle: C = 180 - 72 - 63 = 45°
• Distance from first point to peak: a = 500 x sin(72°) / sin(45°) = 672.50 m
• Distance from second point to peak: b = 500 x sin(63°) / sin(45°) = 630.03 m

Diana determines she is approximately 672.50 meters from the mountain peak at her first observation point. This triangulation technique is the same principle used in GPS systems and historical surveying. For right triangle problems specifically, try our right triangle calculator.

Triangle Properties Reference Table

Tips and Complete Guide

Choosing the Right Input Method

The key to solving any triangle efficiently is selecting the correct input method. If you know all three sides, SSS is the most straightforward method. If you know two sides and the angle between them, SAS is your best option since it directly uses the Law of Cosines without any ambiguity. ASA and AAS are both excellent when you know two angles because the third angle is immediately determined by subtraction. SSA should be used with caution because it can produce the ambiguous case where two valid triangles exist.

The Triangle Inequality Theorem

Before attempting to solve a triangle using the SSS method, verify that your side lengths satisfy the triangle inequality theorem: the sum of any two sides must be greater than the third side. Specifically, a + b > c, a + c > b, and b + c > a must all be true. For example, sides of 3, 4, and 8 cannot form a triangle because 3 + 4 = 7, which is less than 8. If your values fail this check, the calculator will indicate that no valid triangle exists.

Understanding the Ambiguous Case (SSA)

When using the SSA method, be aware of the ambiguous case. Given sides a and b with angle A opposite side a, there are three possible outcomes. If a is less than b x sin(A), no triangle exists. If a equals b x sin(A), exactly one right triangle exists. If a is between b x sin(A) and b (exclusive), two different triangles are possible. If a is greater than or equal to b, exactly one triangle exists. Understanding these conditions helps you interpret results correctly and recognize when multiple solutions may exist.

Applications in Real Life

Triangle calculations appear everywhere in practical applications. Land surveyors use triangulation to map property boundaries. Architects and engineers rely on triangle geometry for structural analysis, roof design, and bridge construction. Navigation systems use triangulation to determine positions. Astronomers use triangulation (parallax) to measure distances to stars. Artists use triangle composition principles for visual balance. Understanding triangle geometry provides a foundation for these and many other fields.

Common Mistakes to Avoid

• Mixing up degrees and radians: This calculator uses degrees. If your textbook gives angles in radians, convert first: degrees = radians x (180 / pi). For example, pi/6 radians = 30 degrees.
• Violating the triangle inequality: If three sides cannot form a closed triangle (the longest side exceeds the sum of the other two), no solution exists. Always verify this when entering side lengths.
• Ignoring the ambiguous SSA case: When using SSA with an acute angle and certain side lengths, two valid triangles may exist. Check whether your problem context limits the solution to one specific triangle.
• Confusing included vs. non-included sides or angles: In SAS, the angle must be between the two known sides. In ASA, the side must be between the two known angles. Using the wrong relationship leads to incorrect results.
• Rounding too early: When solving multi-step triangle problems by hand, keep full precision in intermediate steps. Rounding intermediate values can compound errors significantly in the final result.

Frequently Asked Questions