Quick answer
An isosceles triangle with base b and equal sides l has height h = √(l² − (b/2)²) and area ½bh. Base 6 and legs 5 give a height of 4, area 12, perimeter 16 and base angles of 53.13°.
Enter the base and the equal side. The calculator returns area, height, perimeter and base angle.
Related tools: Triangle Calculator: Area, perimeter, angles and type of a triangle from its three sides. 45-45-90 Triangle Calculator: Sides of a 45°-45°-90° triangle from one leg. 30-60-90 Triangle Calculator: Sides of a 30°-60°-90° triangle from the short leg.
How to use the Isosceles Triangle Calculator
- Enter base (above 0).
- Enter equal side (leg) (above 0).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Base | A number | > 0 |
| Equal side (leg) | A number | > 0 |
Formula and method
Formula
h = √(l² − (b/2)²); A = ½ b h
IEEE-754 double precision; shown to at most 6 decimal places.
The altitude to the base bisects it. Requires l > b/2.
Worked examples
This example is verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Base = 6; Equal side (leg) = 5.
| Result | Value |
|---|---|
| Area | 12 |
| Height | 4 |
| Perimeter | 16 |
| Base angle | 53.130102° |
The first line is the headline result (area). Below it the isosceles Triangle Calculator lists height, perimeter and base angle, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Splitting in half
The height drops from the apex to the middle of the base, forming two right triangles with legs 3 and 4 and hypotenuse 5. The base angle is arccos(3 ÷ 5) = 53.13°, the apex angle 73.74°.
Condition
The legs must be longer than half the base or no triangle exists. If the legs equal the base the triangle is equilateral.
How the result changes with equal side
Holding every other input at the first worked example, this table changes only equal side. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Equal side | Area | Height | Perimeter |
|---|---|---|---|
| 5 | 12 | 4 | 16 |
| 10 | 28.618176 | 9.539392 | 26 |
| 20 | 59.32116 | 19.77372 | 46 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Equal side (leg): entering "5" shows "The equal sides must be longer than half the base." beside that field and no result.
Choosing between the Isosceles Triangle Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Triangle Calculator](/math/triangle-calculator/) | side a, side b and side c | Area |
| [45-45-90 Triangle Calculator](/math/45-45-90-triangle-calculator/) | leg length | Leg |
| [30-60-90 Triangle Calculator](/math/30-60-90-triangle-calculator/) | short leg length | Short leg |
| [Right Triangle Calculator](/math/right-triangle-calculator/) | leg a and leg b | Hypotenuse c |
| [Triangle Area Calculator](/math/triangle-area-calculator/) | base and height | Area |
Background: geometry measurements
Geometry results carry units that depend on the quantity: a perimeter or length is in linear units such as cm, an area or surface area is in square units (cm²) and a volume is in cubic units (cm³). Use one length unit for every input, because the calculator multiplies the numbers as given.
Flat (2D) shapes have an area and a perimeter; solids (3D) have a surface area and a volume. Two familiar checks: a circle of radius r has area πr² and circumference 2πr, and a sphere has volume (4/3)πr³ and surface area 4πr².
Assumptions and limitations
- Legs must be longer than half the base
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for equal side (leg) to confirm this.
Frequently asked questions
How do I find the apex angle?
180° minus twice the base angle.
What if l = b?
The triangle is equilateral.
What is the area formula?
Half base times height.
What does the Isosceles Triangle Calculator calculate?
Height, area and angles of an isosceles triangle from its base and equal sides. It returns Area, Height, Perimeter and Base angle.
What formula does the Isosceles Triangle Calculator use?
h = √(l² − (b/2)²); A = ½ b h. The altitude to the base bisects it.
What do I need to enter in the isosceles Triangle Calculator?
You enter Base and Equal side (leg).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Legs must be longer than half the base.
What is an example of a isosceles Triangle result?
With base 6 and equal side (leg) 5, the area is 12.
Related calculators
- Triangle Calculator — Area, perimeter, angles and type of a triangle from its three sides.
- 45-45-90 Triangle Calculator — Sides of a 45°-45°-90° triangle from one leg.
- 30-60-90 Triangle Calculator — Sides of a 30°-60°-90° triangle from the short leg.
- Right Triangle Calculator — Solve a right triangle from its two legs: hypotenuse, angles, area and perimeter.
- Triangle Area Calculator — Area of a triangle from base and height.
- SAS Triangle Calculator — Solve a triangle from two sides and the included angle.
- Forty-Five Ninety Triangle Calculator — Sides of a 45-45-90 triangle from one leg.
- Triangle Height Calculator — Height of a triangle from its area and base.
- Triangle Perimeter Calculator — Perimeter of a triangle from three sides.
Sources
- OpenStax, Prealgebra 2e (Rice University): perimeter, area, volume, circumference and the Pythagorean theorem
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.