The basic and most common formula for the area of a triangle is:
Area=12×base×height\text{Area}=\frac{1}{2}\times \text{base}\times \text{height}Area=21×base×height
where the base is any side of the triangle, and the height is the perpendicular distance from the base to the opposite vertex
. For the perimeter of a triangle, the formula is simply the sum of the lengths of all three sides:
Perimeter=a+b+c\text{Perimeter}=a+b+cPerimeter=a+b+c
where aaa, bbb, and ccc are the lengths of the sides
. There are also other formulas depending on the known elements:
- Heron's formula for the area when all three sides are known:
s=a+b+c2(semi-perimeter)s=\frac{a+b+c}{2}\quad \text{(semi- perimeter)}s=2a+b+c(semi-perimeter)
Area=s(s−a)(s−b)(s−c)\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}Area=s(s−a)(s−b)(s−c)
This formula allows calculation of the area without knowing the height
- Area using two sides and the included angle :
Area=12×a×b×sinC\text{Area}=\frac{1}{2}\times a\times b\times \sin CArea=21×a×b×sinC
where aaa and bbb are two sides and CCC is the angle between them
- Area of an equilateral triangle (all sides equal):
Area=34×side2\text{Area}=\frac{\sqrt{3}}{4}\times \text{side}^2Area=43×side2
This uses the length of one side
. In summary:
Formula Type| Formula| Conditions/Notes
---|---|---
Area (base and height known)| 12×b×h\frac{1}{2}\times b\times h21×b×h| Base
and perpendicular height known
Perimeter| a+b+ca+b+ca+b+c| Sum of all sides
Area (Heron's formula)|
s(s−a)(s−b)(s−c)\sqrt{s(s-a)(s-b)(s-c)}s(s−a)(s−b)(s−c),
s=a+b+c2s=\frac{a+b+c}{2}s=2a+b+c| All three sides known
Area (two sides and included angle)| 12absinC\frac{1}{2}ab\sin C21absinC|
Two sides and included angle known
Area (equilateral triangle)| 34×side2\frac{\sqrt{3}}{4}\times
\text{side}^243×side2| All sides equal
These formulas cover most cases for calculating the area and perimeter of triangles
