Triangle Calculator – Area, Angles, and Sides
Calculate triangle area, missing sides, and angles using various known measurements.
About This Calculator
Calculate triangle area, missing sides, and angles using various known measurements. Use the calculator above for instant results.
Understanding Triangle Geometry
A triangle is determined only when its side lengths satisfy the triangle inequality: each side must be shorter than the sum of the other two. When all three sides are known, Heron's formula uses the semiperimeter s=(a+b+c)/2 to compute area as √[s(s-a)(s-b)(s-c)]. The law of cosines then recovers the angles from the side lengths.
If only two perpendicular legs of a right triangle are known, the Pythagorean theorem gives c=√(a²+b²). That assumption should be used only when a right angle is actually known. For a general triangle, two arbitrary sides alone are not enough to determine a unique third side and set of angles.
Worked Examples
- Area: 6 | Angles: 36.9°, 53.1°, 90°
Answer: Area=6 sq units
- Area: 10.83 | All angles: 60°
Answer: Area=10.83
- Area: 26.83 | Angles: 46.6°, 57.9°, 75.5°
Answer: Area=26.83
- s=(5+5+6)/2=8.
- Area=√(8·3·3·2)=12.
Answer: area = 12 square units
- Each angle is 60°.
- Area=(√3/4)(4²)=4√3.
Answer: about 6.928 square units
Who Uses This Calculator?
Geometry homework and test prep.
Roof pitch and rafter calculations.
Structural and surveying calculations.
Triangular space and layout design.
Common Mistakes to Avoid
❌ All 3 sides must form a valid triangle
Sum of any two sides must exceed the third. 1-2-10 is impossible (1+2 < 10).
❌ Heron's formula requires all 3 sides
If you only have base and height, area = ½ × base × height instead.
❌ Ignoring the triangle inequality
Sides 2, 3, and 6 cannot form a triangle because 2+3≤6.
❌ Assuming every two-side problem is right-angled
The Pythagorean theorem applies only when the included geometry contains a right angle.
How to Validate Triangle Results
Start with the side lengths. The longest side must be shorter than the sum of the other two; equality would describe a flattened, degenerate triangle with zero area. Once sides are valid, Heron's formula should produce a nonnegative radicand and a positive area for a nondegenerate triangle.
Angles provide a second check. Their total should be approximately 180° after rounding, and the largest angle should lie opposite the longest side. If two sides are equal, the opposite two angles should also be equal. For a right triangle, the longest side c satisfies c²=a²+b² and the angle opposite c is 90°.
Area can be cross-checked when a height or angle is known. The formula A=bh/2 must agree with Heron's formula, and A=(1/2)ab sin C gives another independent route when two sides and their included angle are available. Agreement across formulas is a strong check that the side labels and units were interpreted correctly.
Classification can also verify the geometry. Compare c² with a²+b² using the longest side c: equality indicates a right triangle, c²<a²+b² indicates an acute triangle, and c²>a²+b² indicates an obtuse triangle. This side-only test should agree with the largest computed angle.
When side measurements are rounded, computed angles and area are also approximate. Keep extra digits during intermediate steps and round only the final result so the three angles remain as close as possible to a 180° total.
Frequently Asked Questions
Pythagorean theorem?
For right triangles: a² + b² = c². Where c is the hypotenuse (longest side, opposite the right angle).
What is Heron's formula?
Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2. Works for any triangle with known sides.
How to find angles from sides?
Use the law of cosines: cos(A) = (b²+c²-a²)/(2bc). Then use arccos to find angle A.
What is the triangle inequality?
For a valid nondegenerate triangle, a+b>c, a+c>b, and b+c>a.
Do triangle angles always total 180 degrees?
In ordinary Euclidean plane geometry, yes. A+B+C=180°.
What does Heron's formula require?
It requires all three valid side lengths; no angle or height is needed.
How can I check the computed angles?
Verify that all are positive, each is less than 180°, and their sum is approximately 180° after rounding.
Formula Explorer connections
Interpretation: This relationship connects dimensions, coordinates, angles or trigonometric ratios to a geometric measurement. Assumption: Use a consistent angle mode and length unit, identify the intended shape or coordinate system, and verify that the supplied dimensions form valid geometry.