Scalene Triangle Calculator
Enter three side lengths to calculate area, perimeter, all three interior angles, and the altitude to each side.
Example workbook is ready to download.
Triangle sides
Calculated properties
Angles and altitudes
| Side | Length | Opposite angle | Altitude to side |
|---|---|---|---|
| a | 5 cm | 44.42° | 5.8788 cm |
| b | 6 cm | 57.12° | 4.899 cm |
| c | 7 cm | 78.46° | 4.1991 cm |
Each altitude is the perpendicular distance from a vertex to the line containing the opposite side. Values use the same length unit as the inputs.
How to use this scalene triangle calculator
What this calculator does
This calculator solves a plane triangle from three known side lengths. It checks whether the lengths can form a triangle, then calculates Area, Perimeter, Semiperimeter, the three opposite angles α, β, and γ, and the altitudes to sides a, b, and c. It also reports Triangle by sides and Triangle by angles. The calculation is an exact geometric model of ideal straight sides; it does not account for measurement uncertainty, curved surfaces, or three-dimensional geometry.
When to use it
Use the tool to verify a geometry exercise, check three measured edges before laying out a triangular panel, estimate the surface area of a triangular section, or derive angles and perpendicular heights when only side lengths are available. It also helps identify whether a measured triangle is scalene, isosceles, or equilateral and whether its largest angle is acute, right, or obtuse.
How to calculate
- The calculator opens with a complete demonstration: Side a = 5 cm, Side b = 6 cm, and Side c = 7 cm. The results and a validated example Excel workbook are available immediately.
- Replace Side a, Side b, and Side c with your measurements. Enter ordinary decimal numbers using a period as the decimal separator. Grouped values such as 1,250.5 are accepted; decimal-comma values and scientific notation are rejected to prevent ambiguity.
- Choose Length unit. Switching between millimeters, centimeters, meters, inches, and feet converts the entered lengths so the same physical triangle is preserved. Generic units leave the numbers unchanged.
- Read the live result cards and the Angles and altitudes table. Select Download Excel to create a current-state OOXML workbook with Summary, Inputs, and Geometry sheets.
- Select Reset to clear the demonstration values and calculated content. Reset may disable Download Excel until three complete valid side lengths are entered again.
Input guide
Side a, Side b, and Side c are required positive decimal lengths. Each must be greater than zero and no larger than 10150. A realistic entry might be 5, 6, and 7 centimeters. Increasing one side generally changes every angle, altitude, the perimeter, and the area. The most common mistake is violating the triangle inequality: the sum of any two sides must be greater than the third. Equal sides are accepted, but the classification will correctly show that the triangle is not scalene.
Length unit is required as a display and conversion setting. Choose Generic units, mm, cm, m, in, or ft. The selected unit applies to all side lengths, altitudes, and perimeter; area is shown in the corresponding square unit. Do not mix units across the three side fields. For official guidance on expressing length and area, see NIST's pages on SI length units and SI area units.
Output guide
Area measures the enclosed two-dimensional surface in square units. Perimeter is a + b + c, while Semiperimeter is half that total. A zero result is not produced for a valid triangle; values near zero indicate a nearly degenerate shape. Triangle by sides reports Scalene, Isosceles, or Equilateral. Triangle by angles reports Acute, Right, or Obtuse. Angle sum is the exact plane-triangle identity of 180°.
In the table, Length repeats the canonical input for each side. Opposite angle gives α opposite a, β opposite b, and γ opposite c in degrees. Larger sides face larger angles. Altitude to side gives the perpendicular height associated with that base; it is calculated as 2 × Area divided by the selected side. A long base can therefore have a shorter corresponding altitude even when the triangle's total area is unchanged.
Worked example
For the startup values a = 5 cm, b = 6 cm, and c = 7 cm, the semiperimeter is (5 + 6 + 7) ÷ 2 = 9 cm. Heron's formula gives √(9 × 4 × 3 × 2) = 14.6969 cm². The perimeter is 18 cm. Applying the law of cosines produces α = 44.42°, β = 57.12°, and γ = 78.46°. The three altitudes are approximately 5.8788 cm, 4.8990 cm, and 4.1991 cm. These values match the first rendered results and the startup workbook checkpoints.
For the underlying mathematics, review Heron's formula for area from three sides and the law of cosines for recovering angles.
Formula and interpretation
The calculator first sorts the sides only for the validity test. If x ≤ y ≤ z, a triangle exists only when x + y > z. It then uses the semiperimeter and a numerically scaled form of Heron's formula so large finite inputs are less likely to overflow.
Area = √(s(s – a)(s – b)(s – c))
α = arccos((b² + c² – a²) / (2bc))
hₐ = 2 × Area / a
A nearly flat triangle can be mathematically valid while having a very small area and one angle close to 180°. In measurement work, compare the remaining inequality margin with the precision of your instruments rather than treating a tiny positive area as practically robust.