Triangle Area Calculator
A triangle with base 10 and perpendicular height 6 has an area of 30 square units. This triangle area calculator supports three input sets: base and height, all three side lengths with Heron's formula, or two sides and their included angle with the sine formula. Choose the method that matches known measurements rather than treating the fields as interchangeable. Base and height must be perpendicular. Three sides must satisfy the strict triangle inequality. In the SAS method, the angle must sit between the two entered sides and must be greater than 0 degrees and less than 180 degrees.
Quick answer
The classic formula is area = base x height / 2, half the enclosing rectangle.
Area
30
What this tells you
- •The classic formula is area = base x height / 2, half the enclosing rectangle.
- •Know all three sides but no height? Heron's formula gets the area from the sides alone.
- •Know two sides and the angle between them? Area = a x b x sin(angle) / 2.
- •All three methods give the same answer for the same triangle, so use the one matching your data.
- •The result uses square units, so centimeter inputs produce square centimeters.
- •Heron's method rejects sides when any two add to no more than the third.
- •The SAS method converts degrees to radians before evaluating the sine.
- •The final numeric area is rounded to no more than six decimal places.
How to Use
- 1Choose base-height, three sides, or two sides and included angle based on the measurements available.
- 2Keep every length in one unit. Convert mixed feet and inches or meters and centimeters before calculating.
- 3For base-height, enter the chosen base and the perpendicular distance to the opposite vertex.
- 4For three sides, enter positive lengths a, b, and c that satisfy all three triangle-inequality checks.
- 5For SAS, enter the two positive side lengths and the angle physically between them in degrees.
- 6Calculate and label the answer with squared units, such as cm2, m2, in2, or ft2.
- 7Check the result against a sketch and rough base-height estimate to catch swapped or mistyped inputs.
How It Works
Formula
A = b x h / 2, or A = sqrt(s(s-a)(s-b)(s-c)) with s = (a+b+c)/2For base-height, multiply base by its perpendicular height and divide by 2. For three sides, find semi-perimeter s = (a+b+c)/2, then evaluate sqrt(s(s-a)(s-b)(s-c)). Sides 5, 6, and 7 give s = 9 and sqrt(216) = 14.696938 after rounding. For SAS, convert the included angle from degrees to radians and calculate a x b x sin(C) / 2. With sides 8 and 5 around 60 degrees, the result is 17.320508.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Base 10, height 6
Multiply 10 by the perpendicular height 6 to get 60, then divide by 2 for exactly 30 square units.
Sides 5, 6, and 7
Heron's formula uses s = (5+6+7)/2 = 9. The radicand is 9 x 4 x 3 x 2 = 216, and sqrt(216) is 14.696938 after six-decimal rounding.
Sides 8 and 5 with a 60 degree angle
8 x 5 x sin(60 degrees) / 2 uses sin(60 degrees) = 0.8660254. The result is 17.320508 after rounding.
Right triangle with base 9 and height 12
Multiply 9 x 12 = 108 and divide by 2. The result is exactly 54 square units.
Triangle with sides 13, 14, and 15
The semi-perimeter is 21. Heron's formula gives sqrt(21 x 8 x 7 x 6) = sqrt(7056) = 84.
Sides 7 and 11 with included angle 30 degrees
The sine of 30 degrees is 0.5, so 7 x 11 x 0.5 / 2 equals 19.25 square units.
Triangle Areas for Common Dimensions
Base and height pairs with their areas.
| Base | Height | Area |
|---|---|---|
| 4 | 3 | 6 |
| 10 | 6 | 30 |
| 12 | 8 | 48 |
| 15 | 10 | 75 |
| 20 | 14 | 140 |
How to choose the correct method
Use base-height when a perpendicular altitude is known or easy to measure. The altitude may fall inside the triangle, on a side, or outside an obtuse triangle's drawn boundary. A slanted side is not automatically the height.
Use Heron's formula when all three sides are known but no altitude or angle is available. Check measurement precision carefully for a very thin triangle because small side errors can create a large relative area change.
Use SAS when two sides and the angle between them are known. If the given angle is opposite one side rather than between the entered pair, the direct SAS formula does not match that data set.
Common mistakes
- Using a slanted side as the height. The height is the perpendicular distance from the base to the opposite corner, not the length of a side.
- Forgetting to halve. Base times height gives the surrounding parallelogram, and the triangle is half of it.
- Entering three sides that cannot form a triangle. Sides 1, 2, and 5 fail because 1 + 2 is less than 5.
- Using the wrong angle in the trig method. The angle must sit between the two sides you entered.
- Mixing length units and then labeling the numeric output as though every input used the same unit
- Rounding the sine, semi-perimeter, or intermediate products before the final area
- Using an angle in radians even though the interface expects degrees
- Assuming sides that nearly satisfy the triangle inequality are exact when they may reflect measurement error
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