Triangle Area
Calculate the area of any triangle using base & height or Heron's formula. Supports all triangle types with step-by-step solutions.
What is the Area of a Triangle?
The area of a triangle is half the base times the height (A = ยฝbh). This works for any triangleโscalene, isosceles, or equilateralโas long as the height is perpendicular to the base. When you only know the three side lengths, Heron's formula provides an elegant solution using the semi-perimeter.
Base ร Height Method
A = ยฝ ร b ร hMultiply base by height, then divide by 2. Height must be perpendicular to base.
Heron's Formula
A = โ[s(s-a)(s-b)(s-c)]Where s = (a+b+c)/2 is the semi-perimeter. Works with just the three sides.
Two Sides + Included Angle
A = ยฝ ร a ร b ร sin(C)Use when you know two sides and the angle between them.
Equilateral Triangle
A = (โ3/4) ร sยฒSpecial formula for equilateral triangles where all sides equal s.
Examples
Right Triangle: base 6, height 8
A = ยฝ ร 6 ร 8 = 24 square units = 24
Heron's: sides 3, 4, 5 (right triangle)
s = (3+4+5)/2 = 6. A = โ[6ร3ร2ร1] = โ36 = 6 = 6
Equilateral: side 10
A = (โ3/4) ร 10ยฒ = (โ3/4) ร 100 โ 43.3 = โ 43.3
Scalene: sides 7, 8, 9
s = 12. A = โ[12ร5ร4ร3] = โ720 โ 26.83 = โ 26.83
Frequently Asked Questions
What is the formula for the area of a triangle?
The most common formula is A = ยฝ ร base ร height. The base is any side, and the height is the perpendicular distance from the base to the opposite vertex.
How do I find the area with only 3 sides?
Use Heron's formula: First, find the semi-perimeter s = (a+b+c)/2. Then calculate A = โ[s(s-a)(s-b)(s-c)]. This works for any valid triangle.
What makes sides invalid for a triangle?
The triangle inequality states that the sum of any two sides must be greater than the third. If a + b โค c (for any combination), no triangle can be formed.
Why is the area formula half base times height?
A triangle is exactly half of a parallelogram (or rectangle). If you duplicate a triangle and flip it, you get a parallelogram with area = base ร height. Thus, triangle = ยฝ ร base ร height.
How do I find the height of a triangle?
If you know the area and base: h = 2A/b. For right triangles, one leg is the height. For others, you may need to draw the altitude from a vertex perpendicular to the opposite side.