How do I calculate the area of a triangle?

The most common formula is Area = ½ × base × height. If you know all three sides (a, b, c), use Heron's formula: Area = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2 is the semi-perimeter. The Triangle Calculator supports multiple input methods.

What is the Pythagorean theorem?

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c) equals the sum of the squares of the other two sides: a² + b² = c². For example, a triangle with sides 3 and 4 has a hypotenuse of √(9+16) = 5.

What are the types of triangles?

Triangles are classified by sides: equilateral (all sides equal), isosceles (two sides equal), scalene (all sides different). By angles: acute (all angles 90°).

How do I find a missing angle in a triangle?

The sum of all interior angles in a triangle is always 180°. If you know two angles, subtract their sum from 180° to find the third. For example, if two angles are 45° and 60°, the third angle is 180° - 45° - 60° = 75°.

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Triangle Calculator

Solve any triangle by entering 3 known values (at least one side). Calculates all sides, angles, area, perimeter, heights, medians, inradius, circumradius, and vertex coordinates.

SSSSASASAAASSSADegree & Radian

How to use this calculator
Enter any 3 values (at least one side length). Leave the unknowns blank and click Calculate. Supports Degree and Radian modes. For radian input, you can type expressions like pi/3.

Triangle Calculator

Provide 3 values including at least one side

Angle A

Angle B

Angle C

Side a (opposite A)

Side b (opposite B)

Side c (opposite C)

Angle Unit:Degree °Radian

Results are computed using trigonometric laws (Law of Cosines, Law of Sines).

Triangle Facts & Laws

Interior angles always sum to 180°. The exterior angle equals the sum of the two non-adjacent interior angles.

Triangle inequality: The sum of any two sides must be greater than the third side.

Pythagorean theorem (right triangles only): a² + b² = c², where c is the hypotenuse.

Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). Useful when an angle and its opposite side are known.

Law of Cosines: a² = b² + c² − 2bc·cos(A). Generalizes the Pythagorean theorem to all triangles.

Heron's Formula (area from three sides): Area = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2.

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Tips

• Enter any 3 values (at least 1 side) to solve
• SSS: all 3 sides known
• SAS: 2 sides + included angle
• ASA/AAS: 2 angles + 1 side
• SSA may have 2 solutions (ambiguous case)
• Radian mode accepts expressions like pi/3

In-Depth Guide

The Triangle Calculator is an indispensable online tool designed to simplify complex trigonometric calculations for any triangle. Whether you're dealing with right-angled, acute, or obtuse triangles, this calculator can determine unknown sides, angles, and the area using fundamental trigonometric principles. It works by applying key formulas such as the Law of Sines, the Law of Cosines, and the basic area formula (0.5 * base * height), or Heron's formula for area when all three sides are known.

Understanding how the Triangle Calculator works involves a grasp of basic geometry and trigonometry. For instance, if you input two sides and the included angle (SAS), the calculator uses the Law of Cosines to find the third side, and then the Law of Sines to determine the remaining angles. Similarly, if you provide three sides (SSS), Heron's formula is employed for the area, and the Law of Cosines helps find all angles.

The formulas at the heart of the Triangle Calculator are the bedrock of trigonometry. The Law of Sines states that the ratio of a side's length to the sine of its opposite angle is constant for all three sides and angles in a triangle (a/sin A = b/sin B = c/sin C). The Law of Cosines generalizes the Pythagorean theorem, relating the lengths of the sides of a triangle to the cosine of one of its angles (c² = a² + b² - 2ab cos C).

Beyond academic use, the Triangle Calculator proves invaluable in practical scenarios. Architects might use it to calculate roof pitches or the dimensions of structural components. Surveyors rely on it to determine distances and angles across irregular terrains. In navigation, sailors and pilots use trigonometric principles to plot courses and estimate positions.

Tips & Considerations

→Always double-check your input values to ensure accuracy; even a small error can lead to significantly different results.
→Familiarize yourself with the basic trigonometric laws (Sine, Cosine) to better understand the calculator's output and its implications.
→When dealing with real-world measurements, consider the units used (e.g., meters, feet, degrees, radians) to maintain consistency and correctness.
→For ambiguous cases in the Law of Sines (SSA), be aware that two possible triangles might exist; the calculator typically provides one solution, so verify if another is plausible.
→Use the calculator as a learning aid to reinforce your understanding of geometric principles rather than just a black box solution provider.

Frequently Asked Questions

This Triangle Calculator is versatile and can solve for all types of triangles, including right-angled triangles, acute triangles, and obtuse triangles. It can handle various input combinations, such as Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Side-Side-Side (SSS), and Angle-Angle-Side (AAS). The underlying trigonometric laws are universally applicable to any planar triangle.