To find the area of a triangle, the most common and basic formula is:
Area=12×base×height\text{Area}=\frac{1}{2}\times \text{base}\times \text{height}Area=21×base×height
Here, the base is any side of the triangle, and the height is the perpendicular distance from the opposite vertex to that base
Other methods depending on available information:
- If you know all three sides (a, b, c): Use Heron's formula.
- Calculate the semi-perimeter s=a+b+c2s=\frac{a+b+c}{2}s=2a+b+c.
- Then, the area is:
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 works without needing the height
- If you know two sides and the included angle (SAS): Use the formula:
Area=12×a×b×sin(γ)\text{Area}=\frac{1}{2}\times a\times b\times \sin(\gamma)Area=21×a×b×sin(γ)
where aaa and bbb are the sides, and γ\gamma γ is the angle between them
- For an equilateral triangle (all sides equal to aaa):
Area=34×a2\text{Area}=\frac{\sqrt{3}}{4}\times a^2Area=43×a2
which is approximately 0.433×a20.433\times a^20.433×a2
Summary:
Known Parameters| Formula
---|---
Base and height| 12×base×height\frac{1}{2}\times \text{base}\times
\text{height}21×base×height
Three sides (a, b, c)|
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 (Heron's formula)
Two sides and included angle| 12×a×b×sin(γ)\frac{1}{2}\times a\times b\times
\sin(\gamma)21×a×b×sin(γ)
Equilateral triangle side length| 34×a2\frac{\sqrt{3}}{4}\times a^243×a2
Use the formula that fits the information you have about the triangle to calculate its area accurately
