Sides 3, 4, and 5
Area = 6 · Perimeter = 12
With a = 3, b = 4, c = 5, angle C is exactly 90°. Angles A and B are approximately 36.869898° and 53.130102°. The heights to a, b, and c are 4, 3, and 2.4.
Geometry
Triangle Calculator
Enter three sides, or choose an angle-based input method, to find the sides, angles, area, perimeter, and heights of a triangle.
Enter triangle measurements
Triangle area
Enter the three side lengths, then select Calculate.
Side lengths
- Side a · opposite A
- Side b · opposite B
- Side c · opposite C
- Perimeter
- Angle A · opposite a
- Angle B · opposite b
- Angle C · opposite c
Perpendicular heights
- Height to side a
- Height to side b
- Height to side c
Calculation steps
Your measurements are calculated in your browser. This tool solves flat-plane triangles with SSS, ASA/AAS, or SAS. Angles use degrees; lengths are not converted.
Worked examples
Three sides of length 2
Area = √3 ≈ 1.732051
The perimeter is 6, and every angle is exactly 60°. Each perpendicular height is √3, the same numerical value as the area in this example, but measured in length units instead of square units.
Sides 1, 2, and 3
1 + 2 = 3
The two shorter sides only reach the length of the longest side. They form a straight line with no positive area, so the calculator rejects these lengths. The inequality must be strict.
Optional angle-based examples
A = 30°, B = 60°, c = 10
C = 90° · a = 5
Select Two angles + side and choose c, the side between A and B. The other side is b = 5√3 ≈ 8.660254, the area is ≈ 21.650635, and the perimeter is ≈ 23.660254.
A = 30°, B = 60°, a = 5
Same triangle, different known side
Choose a instead and enter 5, or select Try an example. The sine rule gives c = 10 and b ≈ 8.660254. The known-side selection tells the calculator which angle belongs to your length.
With b selected, Try an example uses A = 60°, B = 30°, and b = 5. The angles exchange labels so b is opposite 30° and its length is exactly 5, avoiding a rounded side length as an input.
a = 3, b = 4, C = 90°
c = 5 · Area = 6
Select Two sides + angle. The angle between the two entered sides is 90°, so the missing side is 5. The perimeter is 12 and the heights to a, b, and c are 4, 3, and 2.4.
Formulas and numerical method
Default SSS method
For sides a, b, and c, the perimeter is P = a + b + c, and the semiperimeter is s = P ÷ 2. Heron's formula gives the area: K = √[s(s − a)(s − b)(s − c)].
The cosine rule gives cos A = (b² + c² − a²) ÷ (2bc), with matching formulas for B and C. The naming and formulas are described in OpenStax's lesson on the law of cosines and Heron's formula. Each angle is reported in degrees. Heights follow from hₐ = 2K ÷ a, with the corresponding side used for each other height.
For numerical stability, the implementation sorts the sides as x ≥ y ≥ z and computes 16K² = [x + (y + z)][z − (x − y)][z + (x − y)][x + (y − z)]. This rearrangement of Heron's formula avoids subtracting a long side from a rounded semiperimeter. Decimal arithmetic uses 120 significant digits, enough to retain this product exactly within the input limits.
Angles use the equivalent expression A = atan2(4K, b² + c² − a²), then convert radians to degrees. This remains useful for very small angles and angles close to 180°. Exact side identities identify 60°, 90°, and 120°; other displayed angles carry an approximation mark.
Area and height displays are rounded by comparing exact integer squares, rather than rounding an already shortened square root. Choose 0, 2, 4, 6, or 10 decimal places, with 6 as the default. Exact halfway values round up. The ≈ mark identifies a rounded area or length, or an approximate angle. Independently rounded angles can sum to slightly more or less than 180°.
Side classifications use the entered values before display rounding. All equal sides are equilateral; exactly two equal sides are isosceles; otherwise the triangle is scalene. Comparing the square of the longest side with the sum of the other two squares identifies an acute, right, or obtuse triangle.
ASA/AAS and SAS methods
In ASA/AAS, C = 180° − A − B and a/sin A = b/sin B = c/sin C. The two entered angles must total less than 180°. Side c is between A and B (ASA); side a or b gives AAS. Three angles alone set the shape but cannot determine its size, so one length is required.
In SAS, C lies between a and b. The missing side follows from c² = a² + b² − 2ab cos C. The calculation uses the equivalent c² = (a − b)² + 4ab sin²(C/2) to avoid subtracting nearly equal values at small angles. Area is K = ab sin C ÷ 2; atan2 expressions give the other angles without the SSA ambiguity.
General angle-based calculations use a separate 120-significant-digit context without rounding intermediate results to the selected display precision. Exact entered values, angle sums, symmetry, and supported special-angle squared values are retained where possible. Rational values and exact square-root ratios use integer midpoint comparisons; other derived values are numerical approximations and carry ≈, even if their displayed digits look simple. Classification uses unrounded input relationships, not displayed angles.
For the geometric relationships, see OpenStax's law of sines and law of cosines. The SSA case, with a non-included angle, may have zero, one, or two solutions and is outside these input methods.
Common questions
Can I solve with two sides and an angle?
Yes. Choose Two sides + angle (SAS) and enter a, b, and the angle C between them. An angle opposite one of the entered sides is SSA, which can have zero, one, or two solutions and is not supported. Three sides (SSS) remains the default.
Can I enter two angles and one side?
Choose Two angles + side (ASA/AAS), enter A and B in degrees, and select the side you know. Choose c if the length is between A and B, or a or b if it is opposite the corresponding angle. Relabel your triangle to match the diagram. Both angles must be positive and their sum must be less than 180°.
Does side c have to be the longest side?
No. Enter the sides in any order. Angle A will be opposite a, angle B opposite b, and angle C opposite c. The calculator sorts a separate copy of the lengths for stable area calculations while preserving your labels in the results.
Why are my three positive lengths rejected?
Positive lengths alone do not guarantee a triangle. The longest side must be strictly shorter than the sum of the other two. For example, 2, 3, 6 cannot meet, and 2, 3, 5 form a straight line. Both are rejected. Enter the intended measurement precision without rounding a barely valid triangle into a degenerate one.
What units should I enter?
Use one length unit throughout and omit its symbol. The calculator performs no conversion. If all sides are in centimeters, perimeter and heights are in centimeters, while area is in square centimeters. Convert mixed units before entering the sides.
What is a height to a side?
It is the perpendicular distance from the opposite vertex to the line containing that side. In an obtuse triangle, the perpendicular can land beyond the side's endpoints, on its extension. A sloping side is not generally the same as a height.
Why might I see an area of 0 or an angle of 180°?
A valid triangle always has positive area and angles strictly between 0° and 180°. Very small measurements or a very thin triangle can round to one of these boundary values at the selected precision. The ≈ mark and result note identify that display rounding. Increase decimal places or use a smaller length unit consistently where useful.
Which number formats are supported?
Each numeric input can have up to 15 digits before the decimal point and 10 after it. Correct thousands grouping such as 1,234.5 is accepted. Use a leading zero for decimals such as 0.5. Zero, negative measurements, fractions, scientific notation, expressions, degree symbols, and unit symbols are rejected. Enter angles as numbers in degrees, strictly between 0 and 180.