Scalene Triangle Area Calculator
Find a triangle's area from base and height, three sides, two sides with the included angle, or two angles with the included side.
Triangle inputs
Choose the information you know, then enter positive measurements.
Live results
Results update as soon as the active inputs form a valid triangle.
The two-dimensional space enclosed by the three sides.
s = (a + b + c) / 2; A = √[s(s – a)(s – b)(s – c)]
Calculation details
The table uses the same canonical values as the result cards and Excel workbook.
| Quantity | Value | Interpretation |
|---|---|---|
| Area | 17.3205 cm² | Space enclosed by the triangle. |
| Perimeter | 20 cm | Total distance around all three sides. |
| Semiperimeter | 10 cm | Half the perimeter; used in Heron's formula. |
| Side set | 5, 7, 8 cm | All three side lengths are different, so the triangle is scalene. |
How to use the scalene triangle area calculator
What this calculator does
This calculator estimates the area enclosed by a triangle using the measurements you already know. It supports four standard geometry methods: Base and height, Three sides (SSS), Two sides and included angle (SAS), and Two angles and included side (ASA). The result is an exact mathematical consequence of the entered measurements, subject only to displayed rounding. It does not verify a physical drawing, surveying setup, or whether a measured height is truly perpendicular.
When to use it
Use the calculator when checking homework, estimating the area of a triangular panel or plot, validating dimensions from a drawing, or comparing two possible triangle layouts. Choose the method that uses measurements you can obtain directly; do not convert an angle into a length or guess a missing side when another supported method already fits your data.
How to calculate
- The calculator opens with a ready-to-use demonstration: Three sides (SSS), cm, and sides 5, 7, and 8. Its workbook is ready immediately.
- Choose a method in Given. Only the inputs required by that formula remain visible.
- Select one Length unit and enter every length in that same unit. Angles are always entered in degrees.
- Read Area first, then use Perimeter, Semiperimeter, Triangle type, Formula used, and the Calculation details table to audit the result.
- Select Download Excel to export the current valid inputs and outputs. Reset clears the demonstration and all entered dimensions; export stays unavailable until a complete valid set is entered again.
Input guide
Given is required and selects the model. Length unit is required for labeling only; changing between physical units converts every populated length field so the same triangle is preserved. Switching to or from generic Units leaves the numeric entries unchanged. In Base and height, enter positive decimal values for Base b and Perpendicular height h; the height must meet the base at 90 degrees. In Three sides (SSS), Side a, Side b, and Side c must be positive and satisfy the triangle inequality, meaning any two sides must add to more than the third. In Two sides and included angle (SAS), both sides are positive lengths and Included angle γ must be greater than 0° and less than 180°. In Two angles and included side (ASA), Included side a is positive, both angles are between 0° and 180°, and Angle β + Angle γ must be less than 180°. Commas are accepted only as thousands separators, so enter 1.5 rather than 1,5. The OpenStax explanation of area provides a useful refresher on square units and the base-height formula.
Output guide
Area measures enclosed two-dimensional space and is shown in the selected square unit. Perimeter is the sum of all three sides; Semiperimeter is half that sum. Those two outputs are not determined from base and height alone. Triangle type reports Scalene when all sides differ, Isosceles when exactly two are equal, Equilateral when all three are equal, or Not determined when the selected inputs do not establish all sides. Formula used identifies the active equation. In the table, Quantity names each metric, Value gives its formatted measurement, and Interpretation explains how to read it. A zero area is not accepted because all supported methods require a nondegenerate triangle.
Worked example
For the startup sides 5 cm, 7 cm, and 8 cm, the semiperimeter is s = (5 + 7 + 8) / 2 = 10 cm. Heron's formula gives A = √[10 × (10 – 5) × (10 – 7) × (10 – 8)] = √300 = 17.3205 cm² after display rounding. The perimeter is 20 cm, and because 5, 7, and 8 are all different, the triangle type is Scalene. These same values appear in the initial result cards, detail table, and downloadable workbook.
How the four area formulas work
Base-height is the most direct method. The perpendicular height represents how far the opposite vertex stands from the chosen base, so the triangle occupies half the area of a rectangle with the same base and height.
When all three sides are known, the calculator uses Heron's formula. First it finds the semiperimeter s, then multiplies the four nonnegative factors inside the square root. See the Wolfram MathWorld treatment of Heron's formula for the standard derivation and notation.
For SAS, the sine of the included angle scales the rectangle-like product of the two known sides. For ASA, the missing angle is 180° – β – γ, then the law of sines determines the remaining sides. The OpenStax non-right triangle chapter covers Heron's formula, the law of cosines, and related triangle-solving methods.
Units, interpretation, and common mistakes
Lengths remain in the selected linear unit, while area is always expressed in the corresponding square unit. For example, centimeter inputs produce square centimeters, not centimeters. NIST's guide to SI units for area explains why the square meter is the SI-derived area unit and how square-unit scaling differs from ordinary length conversion.
- Do not use a slanted side as the height unless it is perpendicular to the selected base.
- For SSS, check the triangle inequality before interpreting the area. A side equal to or longer than the other two combined creates a degenerate or impossible triangle.
- For SAS, use the angle between the two entered sides, not one of the other angles.
- For ASA, keep β + γ below 180°. As the sum approaches 180°, the triangle becomes extremely thin and numerical sensitivity increases.
- Equal sides are mathematically valid, but the result will be labeled Isosceles or Equilateral rather than Scalene.
No chart is shown because this calculator produces one principal scalar result rather than a meaningful multi-point series or part-to-whole breakdown. The detail table is the clearer audit view for the available values.