Last updated: July 31, 2026
Isosceles Triangle Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
h=√(a^2-(b/2)^2); A=(b×h)/2; P=2a+b; V=2×asin((b/2)/a)×180/π; B=(180-V)/2
Where:
- a=equal leg length
- b=base length
- h=height to base
- V=vertex angle(degrees)
- B=base angle(degrees)
Worked Examples
Classic 5-6-5 triangle
Equal legs 5 and base 6 form a neat integer-height triangle.
- 1h=√(5²-3²)=4
- 2A=(6×4)/2=12
- 3P=2×5+6=16
Larger triangle
Use equal legs 10 and base 12 to verify scaling.
- 1h=√(10²-6²)=8
- 2A=(12×8)/2=48
- 3P=20+12=32
Decimal dimensions
A smaller triangle still follows the same formulas.
- 1h=√(7.5²-4.5²)=6
- 2A=(9×6)/2=27
Invalid geometry
If the equal leg is not longer than half the base, no real height exists.
- 1Check a>b/2
- 23 is not greater than 3, so the discriminant is not valid for this calculator.
Introduction
This isosceles triangle calculator solves the full triangle when you know the base and one of the equal legs. It computes the height, area, perimeter, vertex angle, and both base angles from a single consistent set of formulas. The validation step prevents impossible triangles where the altitude would become imaginary.
What this calculator finds
The calculator returns the main geometric properties of an isosceles triangle from side data alone.
Height to the base
Triangle area
Perimeter
Vertex angle at the apex
Each base angle
How the formulas work
Dropping the altitude splits the triangle into two congruent right triangles. That lets you use the Pythagorean theorem for height and inverse sine for the apex angle.
Half-base = b/2
Height uses h=√(a²-(b/2)²)
Area uses A=(b×h)/2
Perimeter uses P=2a+b
Angles follow from sine and the 180° triangle sum
If you already know the height, you can verify the same area result manually.
Input guide
Enter the equal leg as a and the bottom side as b. Both values must use the same unit so the outputs stay consistent.
- 1
a is either equal side
- 2
b is the unique base
- 3
Decimals are allowed
- 4
Use positive lengths only
Validation rules
The equal leg must be longer than half the base, otherwise the altitude formula would require the square root of a negative number.
a must be finite
b must be finite
a>0 and b>0
a>b/2
A value of a=b/2 creates a degenerate straight-line triangle and is rejected here.
Understanding the angles
The vertex angle opens between the equal legs. The two base angles are identical because the legs are equal, so the remaining angle sum is split evenly.
Vertex angle is opposite the base
Base angles are always equal
V+2B=180°
Larger base means larger vertex angle for the same leg length
Where this is useful
This calculator is handy in drafting, carpentry, roof framing, and classroom geometry because side measurements are often easier to collect than angles.
Checking triangle layouts
Estimating material lengths
Reviewing geometry homework
Verifying CAD or drawing dimensions
FAQs
What does this isosceles triangle calculator compute?
It computes the height, area, perimeter, vertex angle, and equal base angles from one equal leg and the base.
Why must the equal leg be greater than half the base?
Because the altitude formula uses √(a²-(b/2)²). If a is not greater than b/2, the quantity inside the square root is zero or negative for this calculator.
What is the vertex angle?
The vertex angle is the apex angle between the two equal sides of the triangle.
Are the base angles always equal?
Yes. In an isosceles triangle, equal legs imply equal base angles.
Can I enter decimals?
Yes. The calculator accepts finite decimal values and rounds answers to 6 decimal places.
What unit should I use?
Any unit works as long as the leg and base use the same unit. Area is then reported in square units.