How to Work Out the Height of a Triangle – Formulas & Examples
How to Find the Height of a Triangle – From Area, Sides or Angles
Triangle height is not a single formula problem. The right method depends on whether you know area and base, a right-triangle relationship, three sides, or a side and angle.
On this page
What Is Triangle Height?
The height (altitude) of a triangle is the perpendicular distance from a chosen base to the opposite vertex. Because any side can be the base, a triangle has three possible heights.
Height from Area and Base
Example: area 54 cm², base 12 cm:
The units also confirm the rearrangement: cm² ÷ cm = cm.
Height in a Right Triangle
If the chosen base is one leg of a right triangle, the other leg is already the perpendicular height. If the hypotenuse c and base b are known, use:
Example: c=13 m and b=5 m → h=√(169−25)=12 m.
Height from Three Sides — Heron’s Formula
With sides a, b and c, first calculate the semiperimeter and area:
A=√[s(s−a)(s−b)(s−c)]
Then choose a base and use h=2A/base.
For sides 13, 14 and 15 cm, choosing base 14:
Height from a Side and Included Angle
If side a makes angle C with base b, the perpendicular component of side a is:
Example: a=10 cm and C=40° → h=10sin40°≈6.428 cm.
This is exactly why the SAS area formula is K=½ab sinC: the term a sinC is the height.
Worked Examples
Area 72 m², base 16 m
Check: ½×16×9=72 m².
Equilateral triangle, side 10 cm
Drop the altitude to split the triangle into two right triangles.
Isosceles triangle, equal sides 5 and base 6
The altitude bisects the base into 3 and 3.
How to Check the Height
Practice Questions
Area 35 cm², base 10 cm
Answer: h=2×35/10=7 cm.
Right triangle: base 9 m, hypotenuse 15 m
Answer: h=√(225−81)=12 m.
Questions People Ask
Open any question for a detailed explanation, not just a one-line answer.
What is the height of a triangle?
The height is the perpendicular distance from a chosen base to the opposite vertex. Each possible base has its own corresponding altitude.
How do I find height from area and base?
Rearrange A=½bh to h=2A/b. Double the area and divide by the base.
Can a sloping side be the height?
Only if it is perpendicular to the chosen base. Height is defined by the 90° relationship, not by which side looks vertical on a drawing.
How do I find height from three sides?
Use Heron’s formula to calculate the area, then use h=2A/base.
What is the height of an equilateral triangle?
For side s, h=(√3/2)s. The formula comes from splitting the equilateral triangle into two 30-60-90 right triangles.
How does sine give triangle height?
In a right-triangle decomposition, sinC=opposite/hypotenuse. If the sloping side is a, the opposite perpendicular component is a sinC, which is the altitude.
Can a triangle’s height fall outside the triangle?
Yes. In an obtuse triangle, an altitude from an acute vertex can meet an extension of the opposite side outside the triangle.
Where can I calculate height automatically?
The Triangle Calculator handles several triangle cases and explains the height relationship in Learn Mode.
Related CalcTypes Resources
Methodology
The guide separates altitude methods by the information given and verifies each height through area, Pythagoras or trigonometric substitution where possible.