Last updated: July 31, 2026
Isosceles Triangle Height Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
h_base=√(b^2-(a/2)^2); h_leg=(a×h_base)/b
Where:
- a=base length
- b=equal leg length
- h_base=height to base
- h_leg=height to equal leg
Worked Examples
5 and 6 example
Equal legs 5 and base 6 create two clean heights.
- 1h_base=√(25-9)=4
- 2h_leg=(6×4)/5=4.8
13 and 10 example
This case gives an integer height to base and a repeating decimal to leg.
- 1h_base=√(169-25)=12
- 2h_leg=(10×12)/13
Decimal dimensions
Both heights remain easy to compute with decimals.
- 1h_base=√(7.5²-4.5²)=6
- 2h_leg=(9×6)/7.5
Invalid dimensions
The equal leg must be longer than half the base.
- 1Check b>a/2
- 23 is not greater than 3, so the altitude to the base is not valid here.
Introduction
This isosceles triangle height calculator returns two different altitudes from the same triangle. One height drops from the apex to the base, and the other drops from a base vertex to one of the equal legs. Because both heights describe the same area, they are linked by a simple area relation.
Why there are two heights
Any side of a triangle can serve as a base, so a triangle can have different corresponding altitudes. This calculator focuses on the two most useful heights in an isosceles triangle.
Height to the unique base
Height to either equal leg
Both represent perpendicular distances
Finding the height to the base
Split the base in half and use the equal leg as the hypotenuse of a right triangle.
Compute a/2
Use h_base=√(b²-(a/2)²)
Round the result to six decimals
Finding the height to a leg
Set the area using the base and the apex altitude equal to the area using a leg and its corresponding altitude.
Area=(1/2)×a×h_base
Area=(1/2)×b×h_leg
Solve h_leg=(a×h_base)/b
This second height is often useful when checking equivalent area formulas.
Input meaning
Here a is the base and b is either equal side. Keep both measurements in the same linear unit.
a is the base
b is an equal leg
Positive decimals are accepted
Validation rules
The equal leg must exceed half the base so the altitude to the base stays real and positive.
b must be finite
a must be finite
b>0 and a>0
b>a/2
If b=a/2, the triangle degenerates and the base altitude becomes zero.
Typical uses
Dual-height calculations are useful in geometry lessons, drafting, truss analysis, and symmetric design work.
Area verification
Construction layouts
Classroom demonstrations
Engineering sanity checks
FAQs
Why does this calculator return two heights?
A triangle can have a different altitude for each side used as a base. This tool reports the altitude to the base and the altitude to an equal leg.
What is height to base?
It is the perpendicular distance from the apex to the base of the isosceles triangle.
What is height to leg?
It is the perpendicular distance from a base vertex to one of the equal legs.
Why is the equal leg compared with half the base?
Because the Pythagorean step uses √(b²-(a/2)²), which requires b to be greater than a/2.
Can I use the heights to verify area?
Yes. The calculator is based on the identity (1/2)×a×h_base=(1/2)×b×h_leg.
Do the outputs use the same unit as the inputs?
Yes. Both heights are linear distances, so they use the same unit as the base and equal leg.