Find the area of any triangle using base-height or Heron's formula.
Two methods: base-height or three sides (Heron\'s formula).
Area
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Formula: ½ × base × height
The Triangle Area Calculator finds the area of any triangle using one of two methods. If you know the base and height, the calculator uses the classic formula Area = ½ × base × height. If you only know the three side lengths, it uses Heron\'s formula, which derives the area without needing the height. The calculator also validates the triangle inequality to ensure the entered sides can actually form a triangle.
Area = ½ × base × height
Here "height" is the perpendicular distance from the base to the opposite vertex. This is the simplest formula and works for any triangle as long as you can identify a base and its corresponding perpendicular height.
s = (a + b + c) ÷ 2
Area = √( s × (s − a) × (s − b) × (s − c) )
Where s is the semi-perimeter. This formula works for any triangle when you know all three side lengths but not the height.
Area = ½ × a × b × sin(C)
Where C is the angle between sides a and b. Useful in trigonometry contexts.
Base-height: A triangle with base 10 and height 6 has area = ½ × 10 × 6 = 30 square units.
Heron\'s formula: For sides 5, 6, 7: s = (5+6+7)/2 = 9. Area = √(9 × 4 × 3 × 2) = √216 ≈ 14.70.
SAS: Two sides 8 and 5 with included angle 60°. Area = ½ × 8 × 5 × sin(60°) = ½ × 8 × 5 × 0.866 = 17.32.
Three side lengths form a valid triangle only if the sum of any two sides exceeds the third. This is the triangle inequality theorem. For sides 3, 4, 5: 3+4 > 5 ✓, 3+5 > 4 ✓, 4+5 > 3 ✓ — valid. For sides 1, 2, 5: 1+2 = 3, which is not greater than 5 — invalid, no triangle exists. The calculator checks this automatically and warns you if the sides cannot form a triangle.
The triangle inequality is fundamental to geometry. It defines what a triangle is — three line segments that can actually meet at three vertices. It also underlies the concept of distance in mathematics: the shortest path between two points is a straight line, so any detour through a third point must be longer. This property generalizes to higher dimensions and is the foundation of metric spaces in advanced mathematics.
Triangle area calculations appear in surveying (measuring irregular land plots by triangulation), architecture (computing roof surfaces), computer graphics (every 3D model is built from triangles), and physics (calculating forces on triangular surfaces). The Pythagorean theorem — perhaps the most famous theorem in mathematics — is essentially a statement about right triangles that connects side lengths to area through the square of the hypotenuse.
The simplest formula is: Area = ½ × base × height. If you only have the three side lengths, use Heron's formula: Area = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2.
Heron's formula computes triangle area from three side lengths alone: Area = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter. It works for any valid triangle.
The triangle inequality theorem: the sum of any two sides must exceed the third. For sides a, b, c: a+b>c, a+c>b, b+c>a. If any fails, no triangle exists.
Semi-perimeter s = (a + b + c) ÷ 2. It is half the perimeter and appears in Heron's formula.
Yes: Area = ½ × a × b × sin(C), where C is the angle between sides a and b. Useful when you have SAS (side-angle-side) information.
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