how to find the area of a triangle

To find the area of a triangle, you can use different formulas depending on what information you have:

1. Using Base and Height

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 (or altitude) is the perpendicular distance from the base to the opposite vertex

2. Using Three Sides (Heron's Formula)

If you know the lengths of all three sides aaa, bbb, and ccc, but not the height, use Heron's formula:

• First, calculate the semi-perimeter:

s=a+b+c2s=\frac{a+b+c}{2}s=2a+b+c​

• Then, compute the area as:

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 for any triangle

3. Using Two Sides and the Included Angle

If you know two sides and the angle between them (say sides aaa and bbb and included angle CCC), use the formula:

Area=12×a×b×sin⁡(C)\text{Area}=\frac{1}{2}\times a\times b\times \sin(C)Area=21​×a×b×sin(C)

This is useful for triangles where height is not known but an angle is given

4. For Equilateral Triangles

If all sides are equal with length aaa, the area can be calculated as:

Area=34a2\text{Area}=\frac{\sqrt{3}}{4}a^2Area=43​​a2

Summary Table of Formulas

Known Parameters| Formula
---|---
Base bbb and Height hhh| 12×b×h\frac{1}{2}\times b\times h21​×b×h
Three sides a,b,ca,b,ca,b,c| s(s−a)(s−b)(s−c)\sqrt{s(s-a)(s-b)(s-c)}s(s−a)(s−b)(s−c)​, where s=a+b+c2s=\frac{a+b+c}{2}s=2a+b+c​
Two sides a,ba,ba,b and angle CCC| 12×a×b×sin⁡(C)\frac{1}{2}\times a\times b\times \sin(C)21​×a×b×sin(C)
Equilateral side aaa| 34a2\frac{\sqrt{3}}{4}a^243​​a2

These formulas cover most cases for finding the area of a triangle based on the available information