How this instrument works
The height of an isosceles triangle is the length of the altitude dropped straight down from the apex — the point where the two equal legs meet — to the base below. In a triangle with no special symmetry, four different lines start from a vertex and land on the opposite side: the altitude (perpendicular to the base), the median (to the base's midpoint), the angle bisector (splitting the vertex angle in half), and the perpendicular bisector of the base itself. An isosceles triangle collapses all four into one single segment, because reflecting the shape across that line maps each equal leg onto the other.
That collapse is exactly what makes the formula so simple. Drop the altitude and it splits the isosceles triangle into two mirror-image right triangles, each with one leg equal to the height, the other leg equal to half the base, and a hypotenuse equal to the original equal side. The Pythagorean theorem then reads leg² = height² + (base⁄2)², and solving for height gives h = √(leg² − (base⁄2)²) — the base gets halved before it's squared, not after.
The formula only returns a real number while the leg is longer than half the base; shrink the leg toward that boundary and the triangle flattens toward a straight line, with the apex sinking closer to the base until height reaches zero and the three points fall onto a single segment. Stretch the leg instead, holding the base fixed, and the triangle grows tall and narrow — a shape closer to a needle than a tent, with the height climbing without any upper limit.
- Enter the triangle's third side into the Base field — the unequal side that sits between the two equal legs.
- Enter the length of either equal side into the Equal leg length field; both legs share this one value by definition.
- Read Height — the length of the perpendicular segment from the apex straight down to the base.
- Change either input and Height recalculates instantly from the same right-triangle split.
Worked example — the 6-5-5 triangle
Take a triangle with base 6 and two equal legs of 5 — a 6-5-5 triangle, roughly the cross-section of a small canvas tent. The altitude from the apex lands exactly on the midpoint of the base, splitting it into two segments of 3 each and creating a right triangle with legs 3 and h and hypotenuse 5. Solve h² = 5² − 3² = 25 − 9 = 16, so h = 4 exactly — the equal leg of 5 is doing double duty as both a flank of the tent and the hypotenuse of the calculation that finds its height.
The same split scales to any valid pair. Widen the base to 8 while holding the legs at 5 and half the base becomes 4, so h² = 5² − 4² = 9 and h = 3 — a 3-4-5 triple read the other way round. Stretch the legs to 13 with a base of 10 and half the base is 5, landing on the 5-12-13 triple for a height of exactly 12. Every answer traces back to how the leg compares with half the base, the same comparison that keeps the shape from flattening out.
Questions
How is the height of an isosceles triangle found from its base and legs?
It comes straight from the Pythagorean theorem applied to half the triangle: h = √(leg² − (base⁄2)²). Dropping the altitude from the apex splits the shape into two right triangles, each with legs h and base⁄2 and hypotenuse equal to the original leg, so solving that right triangle for its missing side gives the height directly.
Why does one line act as the altitude, median, and angle bisector all at once?
Because an isosceles triangle is symmetric across the line through its apex and the midpoint of its base — folding the shape along that line maps each equal leg exactly onto the other. Any line with that much symmetry has to be perpendicular to the base, pass through its midpoint, and split the apex angle evenly, so the altitude, median, and angle bisector are forced to coincide. A scalene triangle has no such symmetry, so its three equivalents land at three different points.
What's the most common mistake when computing this height?
Squaring the full base instead of half of it. The formula needs base⁄2 — the run of the small right triangle formed by the altitude — not the whole base, so skipping that halving step inflates the value under the square root and produces a height that's too short, or in extreme cases makes the expression negative and undefined.
What happens if the leg is barely longer than half the base?
The triangle flattens toward a straight line. As leg approaches base⁄2, the quantity under the square root shrinks toward zero, the apex sinks toward the base, and height drops toward zero along with it. At leg = base⁄2 exactly, the three vertices become collinear and no triangle exists at all, which is why a valid isosceles triangle always needs the leg strictly longer than half the base.
How does this height relate to the triangle's area?
It's the missing piece: area = ½ × base × height, so once this sheet returns the height, multiplying by half the base finishes the job. This calculator reports only the altitude itself — a separate isosceles triangle area sheet carries that same right-triangle split one step further and multiplies through for you.