Triangle Area Calculator
Find a triangle's area from base and height, three sides, two sides with the included angle, or one side with two adjacent angles.
Triangle measurements
Choose what you know, keep every length in the same unit, and the area updates automatically.
Select the formula that matches the measurements available.
Changing units converts all entered lengths and the squared area unit.
Required positive length for the chosen base.
Required altitude measured at 90° to the base.
Required first side length.
Required second side length.
Required third side length; all three must satisfy the triangle inequality.
Required positive length forming one side of the included angle.
Required positive length forming the other side of the included angle.
Required interior angle between the two entered sides; greater than 0° and less than 180°.
Required side lying between the two entered angles.
Required adjacent interior angle greater than 0°.
Required adjacent interior angle; β + γ must remain below 180°.
Calculated area
Triangle area
45 in²
Using a base of 12 in and a perpendicular height of 7.5 in.
Formula used
A = ½bh
Area unit
in²
Triangle area is 45 square inches.
Calculation details
The table shows the canonical values used in the active formula.
| Item | Value | Unit |
|---|---|---|
| Base (b) | 12 | in |
| Perpendicular height (h) | 7.5 | in |
| Coefficient | 0.5 | ratio |
| Triangle area | 45 | in² |
All length inputs are interpreted in the selected unit. Area is therefore reported in that unit squared.
How to use the Triangle Area Calculator
What this calculator does
This calculator estimates the area enclosed by a valid plane triangle. It supports four standard data sets: Base and perpendicular height, Three sides (SSS), Two sides and included angle (SAS), and One side and two adjacent angles (ASA). The result is an exact numerical consequence of the entered measurements and the selected formula, subject only to display rounding. It does not verify a physical drawing, measurement accuracy, surface slope, material waste, or whether a real-world site is perfectly triangular.
When to use it
Use it to check geometry homework, estimate the face area of a triangular component, calculate a triangular plot from field measurements, or compare alternative measurement methods when a perpendicular height is difficult to obtain. The base-and-height method is usually simplest; SSS is useful when all three edges are known; SAS is convenient when two edges and their included angle are measured; and ASA works when one side and the two angles at its endpoints are known.
How to calculate
- The calculator opens with a ready-to-use demonstration: a 12 in base and a 7.5 in perpendicular height. Its XLSX workbook is immediately available through Download Excel.
- Choose a Calculation method that matches the facts you have. Only the fields required by that method are shown.
- Select the Length unit. When you change it, the calculator converts all stored length values; the physical triangle and its area remain equivalent.
- Replace the demonstration measurements. Results, the formula card, summary pills, and calculation-detail table update live.
- Read Triangle area, confirm the Formula used and Area unit, then export the current canonical values with Download Excel.
- Reset clears all demonstration and user-entered measurements, restores a neutral base-and-height method in centimeters, removes calculated content, and may disable Download Excel until a complete valid state is entered again.
Input guide
Calculation method is required and selects the mathematical model. Length unit is required and accepts mm, cm, m, in, ft, or yd; every entered length must use that same unit. Length fields accept positive decimal numbers using a period as the decimal separator, with optional correctly grouped thousands such as 1,250.5. Ambiguous decimal-comma values such as 1,5, scientific notation, zero, negatives, and values above the supported limit are rejected.
For Base and perpendicular height, enter Base (b) and Perpendicular height (h); both are required positive lengths. The opening example is 12 in and 7.5 in. Increasing either value increases area in direct proportion. The common mistake is using a sloping side as the height – the height must meet the base at 90°. OpenStax's explanation of the base-and-height area formula illustrates this relationship.
For Three sides (SSS), Side a, Side b, and Side c are all required positive lengths; 13, 14, and 15 are a useful example. The sum of any two sides must be greater than the third. Increasing one side does not always increase area because the shape also changes, and a nearly degenerate triangle can have an area close to zero. The method uses Heron's formula, discussed in OpenStax's non-right triangle and Heron formula lesson.
For Two sides and included angle (SAS), First side (a) and Second side (b) are required positive lengths, while Included angle (γ) is required in degrees strictly between 0° and 180°. A realistic example uses 9 in, 5 in, and 30°. The angle must be the angle between the two entered sides; using a non-included angle is a frequent interpretation error. Area rises with either side and follows the sine of the angle, reaching its largest value for fixed sides at 90°.
For One side and two adjacent angles (ASA), enter the required positive Included side (a), then Angle β and Angle γ in degrees. A practical example is 10 cm, 50°, and 60°. Each angle must be greater than 0°, and their sum must be less than 180° so the remaining angle is positive. The side must lie between the two entered angles. An angle sum of 180° or more cannot form a triangle.
Output guide
The Method pill repeats the active formula family, the Unit pill shows the current length unit, and the Status pill reports Ready, Incomplete, or Check inputs. Triangle area is the primary output in square units. A zero area is not produced for a valid nondegenerate triangle; a very small positive result usually means a very small altitude, a very acute or obtuse SAS angle, or SSS sides close to the triangle-inequality boundary. Formula used identifies the identity applied, and Area unit confirms that length conversion is squared. The Calculation details table has Item, Value, and Unit columns and exposes the active inputs plus important intermediates such as semiperimeter, sine, or the third angle. These outputs are exact formula results based on your measurements, not recommendations.
Worked example
The startup example selects Base and perpendicular height with b = 12 in and h = 7.5 in. The calculator applies A = ½bh. The operation is 0.5 × 12 × 7.5 = 45. The first-open result is therefore 45 in², and the same typed values appear in the detail table and exported workbook.
How the formulas work
The most direct identity is one-half of base times perpendicular height. It follows from pairing two congruent triangles to form a parallelogram. When height is unavailable, the other methods derive the same geometric area from different observations.
In the SSS expression, s = (a + b + c) / 2 is the semiperimeter. In SAS, the sine term converts one side into the perpendicular component needed for height. In ASA, trigonometric relationships reconstruct the missing scale information from one side and two angles. Wolfram MathWorld provides a deeper treatment of Heron's formula and equivalent triangle-area identities.
Units, interpretation, and common mistakes
Area uses squared units because it measures two-dimensional extent. Converting a length by a factor converts area by the square of that factor: 1 m equals 100 cm, but 1 m² equals 10,000 cm². The calculator performs length conversion before recalculating so the represented physical area remains unchanged. NIST explains both the SI treatment of area and square units and the dimensional reasoning behind measurement-unit conversion.
Keep all source measurements in one unit, identify the included angle carefully, and do not round intermediate values before the final result. For SSS, verify the triangle inequality before trusting a drawing. For base-height, confirm perpendicularity. For ASA, remember that angles alone do not determine area; one side is necessary to establish scale. Downloaded workbooks preserve canonical numeric values rather than copying rounded display text.