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
