Triangles are one of the most fundamental shapes in geometry, and calculating their properties — sides, angles, perimeter, and area — is a common task in math, science, engineering, and everyday life. Our free online triangle calculator lets you input any known values and instantly computes all the missing measurements. Whether you're dealing with right triangles, equilateral triangles, or any other type, this tool handles the complex trigonometry for you.
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A triangle is a polygon with three sides and three angles. The sum of all three interior angles always equals 180 degrees. Triangles can be classified by their sides or by their angles.
By sides: Equilateral triangles have all three sides equal (and all angles are 60°). Isosceles triangles have two equal sides and two equal angles. Scalene triangles have all sides and angles different.
By angles: Acute triangles have all angles less than 90°. Right triangles have one angle exactly 90°. Obtuse triangles have one angle greater than 90°.
Understanding these classifications helps you choose the right formulas and approach for solving any triangle problem. The most commonly calculated properties include the three side lengths, three angle measures, perimeter, area, and height.
Our triangle calculator is flexible and works with various input combinations. Here's how to use it effectively:
For Right Triangles: If you know any two sides, enter them and the calculator uses the Pythagorean theorem (a² + b² = c²) to find the third side. It also computes both acute angles using inverse trigonometric functions.
For Any Triangle (SSS): Enter all three side lengths. The calculator uses the Law of Cosines to find all three angles. This is useful when you've measured the sides of a triangular plot of land or a physical object.
For Any Triangle (SAS): Enter two sides and the included angle. The calculator uses the Law of Cosines to find the third side, then the Law of Sines for the remaining angles.
For Any Triangle (ASA or AAS): Enter two angles and one side. The calculator finds the third angle (since angles sum to 180°) and uses the Law of Sines to determine the remaining sides.
No matter which values you start with, the calculator outputs all sides, all angles, the perimeter, the area, and the heights — giving you a complete picture of the triangle.
Problem: A right triangle has legs of 5 cm and 12 cm. Find the hypotenuse and area.
Step 1: Apply the Pythagorean theorem: c² = 5² + 12² = 25 + 144 = 169.
Step 2: Take the square root: c = √169 = 13 cm.
Step 3: Calculate area: A = (5 × 12) / 2 = 30 cm².
Result: Hypotenuse = 13 cm, Area = 30 cm².
Problem: A triangle has sides 7, 8, and 9. Find all three angles.
Step 1: Use the Law of Cosines for angle A (opposite side 7): cos(A) = (8² + 9² − 7²) / (2 × 8 × 9) = (64 + 81 − 49) / 144 = 96/144 = 0.6667.
Step 2: A = arccos(0.6667) ≈ 48.19°.
Step 3: For angle B (opposite side 8): cos(B) = (7² + 9² − 8²) / (2 × 7 × 9) = (49 + 81 − 64) / 126 = 66/126 ≈ 0.5238.
Step 4: B = arccos(0.5238) ≈ 58.41°.
Step 5: C = 180° − 48.19° − 58.41° = 73.40°.
Result: Angles are approximately 48.19°, 58.41°, and 73.40°.
Problem: A triangle has angles 40° and 65° with a side of 10 opposite the 40° angle. Find the remaining side.
Step 1: Find the third angle: C = 180° − 40° − 65° = 75°.
Step 2: Use the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). We know a = 10 and A = 40°.
Step 3: Find side b (opposite 65°): b = 10 × sin(65°) / sin(40°) = 10 × 0.9063 / 0.6428 ≈ 14.10.
Step 4: Find side c (opposite 75°): c = 10 × sin(75°) / sin(40°) = 10 × 0.9659 / 0.6428 ≈ 15.03.
Result: Sides are approximately 10, 14.10, and 15.03.
Problem: Find the area of a triangle with sides 13, 14, and 15.
Step 1: Calculate the semi-perimeter: s = (13 + 14 + 15) / 2 = 21.
Step 2: Apply Heron's formula: A = √(s(s−a)(s−b)(s−c)) = √(21 × 8 × 7 × 6) = √7056 = 84.
Result: Area = 84 square units.
Builders use triangle calculations constantly. Roof pitches, stair angles, and structural supports all involve right triangles. Knowing how to calculate angles and sides ensures that structures are built safely and to specification. A triangle calculator eliminates manual computation errors that could compromise structural integrity.
Property boundaries are often defined by triangular sections. Surveyors measure distances and angles to map out plots of land. Triangle calculators help determine the area of irregularly shaped properties by breaking them into triangles, a technique that's been used since ancient Egyptian times.
Triangulation is the principle behind GPS technology. By measuring distances from multiple satellites, a GPS receiver can determine its exact position. Understanding triangle math helps in navigation, whether you're plotting a course at sea or designing a wireless network.
Trigonometry and triangle geometry are core topics in high school and college mathematics. Physics, engineering, and computer graphics all rely heavily on triangle calculations. Students use triangle calculators to verify their work and build intuition for geometric relationships.
Here are the essential formulas every triangle calculator uses:
Pythagorean Theorem (right triangles only): a² + b² = c², where c is the hypotenuse.
Area (general): A = ½ × base × height, or use Heron's formula: A = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2.
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C).
Law of Cosines: c² = a² + b² − 2ab·cos(C).
Perimeter: P = a + b + c.
You need at least three pieces of information, with at least one being a side length. Valid combinations include: SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side), AAS (two angles and a non-included side), and for right triangles, any two sides or one side and one acute angle. Three angles alone (AAA) is not sufficient because it only determines the shape, not the size.
Absolutely. Our calculator handles all triangle types — acute, right, and obtuse. The same trigonometric formulas apply to all triangles. The only difference is that obtuse triangles have one angle greater than 90°, which affects which trigonometric function values are positive or negative during the calculation.
The Law of Sines relates sides to their opposite angles and is most useful when you know an angle-side pair (ASA, AAS, or SSA cases). The Law of Cosines is a generalization of the Pythagorean theorem and works best when you know all three sides (SSS) or two sides and their included angle (SAS). In practice, both laws are used together to fully solve most non-right triangles.
Our calculator provides results with up to 10 decimal places of precision. For practical applications like construction or land surveying, you can round the results to the precision of your input measurements. The underlying calculations use standard trigonometric functions that are accurate to within floating-point precision limits.
Yes. Once the calculator has all sides and the area, it can compute the height relative to any side using the formula h = 2A/base. The calculator displays heights for all three sides, which is useful for engineering and architectural applications where different reference heights may be needed.
Dive deeper with our Area Calculator Guide or learn about proportional reasoning in our Ratio Calculator Guide.