To find the height of a triangle without knowing its area, you can use several methods depending on what information you have about the triangle:

1. Using Side Lengths and Heron's Formula (for any triangle)

• Calculate the semi-perimeter s=a+b+c2s=\frac{a+b+c}{2}s=2a+b+c​, where a,b,ca,b,ca,b,c are the side lengths.
• Find the area A=s(s−a)(s−b)(s−c)A=\sqrt{s(s-a)(s-b)(s-c)}A=s(s−a)(s−b)(s−c)​ using Heron's formula.
• Then, find the height hhh relative to a chosen base bbb by h=2Abh=\frac{2A}{b}h=b2A​.

2. Using Pythagorean Theorem (for right or isosceles triangles)

• For a right triangle, if you know the two legs aaa and bbb, the height relative to the hypotenuse ccc is h=abch=\frac{ab}{c}h=cab​.
• For an isosceles triangle with two equal sides aaa and base bbb, height is h=a2−b24h=\sqrt{a^2-\frac{b^2}{4}}h=a2−4b2​​.

3. Using Trigonometry (if you know two sides and the included angle)

• If you know side bbb and the angle AAA opposite the height, height can be found by:

h=b×sin⁡(A)h=b\times \sin(A)h=b×sin(A)

• Alternatively, if you know two sides a,ca,ca,c and the included angle β\beta β, the height relative to base bbb can be found using:

h=a×sin⁡(γ)h=a\times \sin(\gamma)h=a×sin(γ)

4. Special Case: Equilateral Triangle

• If all sides are equal with length aaa, height is:

h=a32h=\frac{a\sqrt{3}}{2}h=2a3​​

Summary Table

Triangle Type| Known Info| Height Formula
---|---|---
Any triangle| Sides a,b,ca,b,ca,b,c| Use Heron's formula then h=2Abh=\frac{2A}{b}h=b2A​
Right triangle| Legs a,ba,ba,b, hypotenuse ccc| h=abch=\frac{ab}{c}h=cab​
Isosceles triangle| Equal sides aaa, base bbb| h=a2−b24h=\sqrt{a^2-\frac{b^2}{4}}h=a2−4b2​​
Equilateral triangle| Side aaa| h=a32h=\frac{a\sqrt{3}}{2}h=2a3​​
Any triangle| Side bbb, angle AAA| h=bsin⁡(A)h=b\sin(A)h=bsin(A)

These methods allow you to find the height without directly knowing the area, by either calculating the area first or using geometric and trigonometric properties