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.
Siding for a gable end on a 6:12 roof
Luis is pricing siding for the triangular gable end of a 28-foot-wide garage with a 6:12 roof pitch. The roof rises 6 inches for every foot of run, and the run from the wall to the ridge is half of 28, or 14 feet, so the peak sits 14 x 6/12 = 7 feet above the top of the wall. Entering base 28 and height 7 gives 98 square feet of siding before the waste allowance his supplier recommends.
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.
Which triangle area formula to use
Pick the formula by what you can measure. Every row is the same triangle with sides 7, 8, and 9, so the matching areas show that the three methods agree once the inputs describe the same shape.
| What you know | Formula and who it is attributed to | Pick in the calculator | Area of the 7, 8, 9 triangle |
|---|---|---|---|
| Base 9 and its perpendicular height 5.962848 | A = b x h / 2 (Euclid, Elements, Book I, Proposition 41) | Base and height | 26.832816 |
| Base 8 and its perpendicular height 6.708204 | A = b x h / 2, any side can serve as the base | Base and height | 26.832816 |
| Base 7 and its perpendicular height 7.666519 | A = b x h / 2 | Base and height | 26.832816 |
| All three sides, no height or angle | A = sqrt(s(s-a)(s-b)(s-c)) with s = (a+b+c)/2 (Heron of Alexandria, Metrica, first century) | Three sides (Heron) | 26.832816 |
| Sides 7 and 8 and the 73.3985 degree angle between them | A = a x b x sin(C) / 2, the base-height rule with h = b x sin(C) | Two sides and included angle | 26.832823 |
| Sides 8 and 9 and the 48.1897 degree angle between them | A = a x b x sin(C) / 2 | Two sides and included angle | 26.832822 |
| Sides 7 and 9 and the 58.4119 degree angle between them | A = a x b x sin(C) / 2 | Two sides and included angle | 26.832826 |
| Two legs of a right triangle | A = leg x leg / 2, a base-height case because the legs meet at 90 degrees | Base and height, with the legs as base and height | Not applicable. 7 squared plus 8 squared is 113, not 81, so this triangle has no right angle |
Computed with the calculator above and rounded to six decimals. The heights and angles were derived from the 7, 8, 9 sides and entered to six and four decimals, which is why the angle rows land a few millionths above 26.832816. Sources: Heron of Alexandria, Metrica, Book I (first century) for the three-sides formula, and Euclid, Elements, Book I, Proposition 41 for the base-height rule. The coordinate (shoelace) formula of Meister (1769) and Gauss is not offered in this tool.
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. Heron's own worked example in the Metrica used sides 13, 14, and 15, the triangle with area 84 in the examples above. 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.
Three lengths only make a triangle when each one is shorter than the other two combined, the triangle inequality that Euclid proves as Elements, Book I, Proposition 20. Heron's formula enforces it by itself. For sides 1, 2, and 5, the semi-perimeter is 4 and the radicand is 4 x 3 x 2 x (-1) = -24, so the square root is imaginary and no real triangle exists. This calculator runs the inequality check first and shows no result instead of an error value. Watch the near misses too. Sides 10, 10, and 19.99 pass the check and give an area of 3.160301, while 10, 10, and 19.999 give 0.999938, so a measurement error of a hundredth of a unit changes the area by a factor of three. Kahan's note Miscalculating Area and Angles of a Needle-like Triangle (1986) works through why Heron's formula loses accuracy on thin triangles like these.
The angle field expects degrees. The calculator multiplies the entry by pi/180 before taking the sine, the conversion that defines the degree as a unit accepted for use with the SI (NIST Special Publication 811). Entering 1.0472, the radian measure of 60 degrees, is read as 1.0472 degrees. With sides 8 and 5 that gives an area of 0.365521 instead of 17.320508, about 47 times too small. The angle must also sit strictly between 0 and 180 degrees, and it must be the angle between the two sides you entered, not the angle opposite one of them.
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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