Case 1
- Base
- 6
- Equal side (leg)
- 5
12.0000
Open with these valuesIsosceles Triangle Calculator
12.0000units²
Result: 12.0000 units²A perpendicular from the apex splits the triangle into two right halves, so the height is √(leg² − (base ÷ 2)²) and the area ½ × base × height. A base of 6 with legs of 5 gives a height of 4 and an area of 12. The leg has to be longer than half the base.
The numbers at a glance
Held fixed: Base 6.000.
| Equal side (leg) | Result |
|---|---|
| 2.000 | 0.0000 |
| 4.000 | 7.9373 |
| 5.000Your value | 12.0000 |
| 6.000 | 15.5885 |
| 8.000 | 22.2486 |
| 10.000 | 28.6182 |
Worked examples
Case 2
- Base
- 10
- Equal side (leg)
- 13
60.0000
Open with these valuesCase 3
- Base
- 4
- Equal side (leg)
- 4
6.9282
Open with these valuesHow it's calculated
A = ½ × base × √(leg² − (base ÷ 2)²)
- StepMeasure the base — the side the two equal legs stand on.
- StepMeasure one of the two equal legs, in the same unit.
- ResultRead the area; height and both angles follow from the same pair.
What this number means
Enter the base and one leg, not two legs
The two fields want the odd side out — the base the equal legs stand on — and the length of one of those legs. Both in the same unit, and the area comes back in that unit squared.
Two equal sides mean two equal base angles
The equal legs force the symmetry: both base angles come out as acos((base ÷ 2) ÷ leg), and the apex takes whatever is left of 180°. A base of 6 with legs of 5 gives 53.13° twice and 73.74° at the top.
The leg must be longer than half the base
Otherwise the legs cannot meet above the base, the shape flattens into a line and the area is zero. A leg of 2 on a base of 6 is exactly that case, and zero there is the true value rather than an error.
Same area, different triangle
Base 6 with leg 5 and base 8 with leg 5 both cover 12 square units, from heights of 4 and 3. The area alone does not pin down the shape.
Commonly misread
The leg is the height of the triangle.
The leg is a slanted side; the height is the perpendicular onto the base, √(leg² − (base ÷ 2)²). For a leg of 5 on a base of 6 that is 4, not 5, and the height is always the shorter of the two.
The apex angle equals the base angle.
Only in the equilateral case — base 4 with leg 4, where all three are 60°. Otherwise the apex is 180° minus twice the base angle: 73.74° against 53.13° for base 6 and leg 5.
An area of zero means the calculator failed.
It means the leg cannot span half the base, so the triangle has collapsed into a line. A line has no area, and zero is the honest answer.
Reference table
| Base, leg | Height · perimeter · base angle | Area |
|---|---|---|
| 2, 10 | 9.9499 · 22 · 84.26° | 9.9499 |
| 4, 4 | 3.4641 · 12 · 60° | 6.9282 |
| 6, 5 | 4 · 16 · 53.13° | 12.0000 |
| 8, 5 | 3 · 18 · 36.87° | 12.0000 |
| 10, 13 | 12 · 36 · 67.38° | 60.0000 |
Questions
How do I find the area of an isosceles triangle?
First find the height by dropping a perpendicular onto the base: height = √(leg² − (base ÷ 2)²). Then the area is ½ × base × height. A base of 6 with legs of 5 gives a height of 4 and an area of 12 square units.
How do I find the height of an isosceles triangle?
The height is the perpendicular from the apex to the middle of the base. That line splits the shape into two right triangles, so height = √(leg² − (base ÷ 2)²). For a leg of 5 and a base of 6 it is √(25 − 9) = 4, and the height is always shorter than the leg.
What are the base angle and the apex angle?
The two base angles sit where the equal legs meet the base and are equal by symmetry, each acos((base ÷ 2) ÷ leg). The apex angle is at the top between the legs and equals 180° − 2 × base angle. For base 6 and leg 5 that is 53.13° twice and 73.74° at the apex.
Why does a short leg give an area of zero?
An isosceles triangle only closes when the leg is longer than half the base, so the two legs can meet above it. A leg of 2 on a base of 6 cannot reach, the shape flattens into a line, and a flat shape has no area. Zero is the honest answer, not an error.
What units does the calculator use?
The lengths are unit-agnostic: enter base and leg in centimetres, metres or inches and the area comes back in that unit squared. Height and perimeter stay in the entered unit, and the three angles are always in degrees.
Sources and last check
Information, not professional advice.