Triangle Area Calculator

Find a triangle's area from its base and height (A = ½ × base × height) or from three side lengths using Heron's formula.

Quick Facts

Base & height formula
A = ½ × base × height
The height must be measured perpendicular to the chosen base.
Heron's formula
A = √(s(s−a)(s−b)(s−c))
s is the semiperimeter, (a + b + c) / 2, using all three side lengths.
Triangle inequality
a + b > c (for every pair)
Three lengths only form a real triangle if this holds for all three pairings.

Your Results

Calculated
Area
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Triangle area in square units
Perimeter
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Sum of the three sides
Semiperimeter
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s = perimeter ÷ 2
Formula Used
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Method applied to your inputs

Ready

Choose a method, enter your measurements, then press Calculate.

Formula and Method for Triangle Area

Base & Height:

A = ½ × base × height

or A = √(s(s−a)(s−b)(s−c)) using Heron's formula when only the three sides are known

A triangle's area can be found two standard ways, and this calculator supports both. If you know one side (the base) and the perpendicular height to that side, use A = ½ × base × height. If you only know the lengths of all three sides, use Heron's formula: compute the semiperimeter s = (a + b + c) / 2, then A = √(s(s−a)(s−b)(s−c)).

How the calculation works

For the base-and-height method, the calculator simply multiplies the base by the height and halves the result — this works for any triangle (acute, right, or obtuse) as long as the height is measured perpendicular to the base, not along one of the other sides. For the three-sides method, the calculator first checks the triangle inequality (the sum of any two sides must exceed the third), then applies Heron's formula: it finds the semiperimeter s, multiplies (s−a), (s−b), and (s−c) together with s, and takes the square root. Enter all measurements in the same unit — mixing feet and inches, or centimeters and meters, produces incorrect results.

Common mistakes

  • Using the slant side instead of the height: the height must be the perpendicular (90°) distance from the base to the opposite vertex, not the length of an adjacent side.
  • Units: area is in square units (ft², m²). A triangle with a 10 ft base and 6 ft height has an area of 30 ft², not 30 ft.
  • Invalid side lengths: for the three-sides method, if the two shorter sides don't add up to more than the longest side, no triangle exists and Heron's formula cannot produce a real result.

Real-world applications

  • Roofing, siding, and paint estimates often break a surface into triangular sections to calculate material needed (add 10–15% for waste and cuts).
  • Land surveying and real estate use the three-sides method (Heron's formula) when only boundary lengths are measured, not perpendicular distances.
  • Engineering and construction use triangle area for structural gussets, trusses, and load-bearing calculations.
  • Trigonometry and geometry coursework use both methods to connect the concepts of height, side length, and the semiperimeter.

Frequently Asked Questions

What is the formula for the area of a triangle?
The most common formula is A = ½ × base × height, where the height is measured perpendicular to the chosen base. For example, a triangle with a base of 10 ft and a height of 6 ft has an area of ½ × 10 × 6 = 30 ft².
How do I find the area of a triangle from three side lengths?
Use Heron's formula: compute the semiperimeter s = (a + b + c) / 2, then the area is A = √(s(s−a)(s−b)(s−c)). This works for any triangle when you know all three side lengths but not the height.
Why does the calculator say my three sides are invalid?
Three lengths only form a triangle if the sum of any two sides is greater than the third side (the triangle inequality). If a + b ≤ c for any arrangement, no triangle can be built from those lengths, and Heron's formula would require the square root of a negative number.
Does it matter which side I use as the base?
No. Any side can serve as the base as long as you measure the height perpendicular to that same side. Every choice of base and matching height gives the same area.

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