Isosceles Triangle Calculator
Enter any 2 of 8 values — leg a, base b, either altitude (hb or ha), vertex or base angle, area, or perimeter — and get every remaining side, angle, altitude, area, perimeter, inradius and circumradius, with a to-scale diagram and full working.
What is an isosceles triangle?
An isosceles triangle is a triangle with two sides of the same length. Those two equal sides are called the legs; the third side is the base. The mirror symmetry between the two legs forces the two angles at the base to be equal as well — this is the isosceles triangle theorem: equal sides face equal angles, and (the converse) equal angles face equal sides.
Because of that symmetry every isosceles triangle has an axis of symmetry: the perpendicular from the apex to the midpoint of the base. That single line is the altitude to the base, the median to the base, the angle bisector of the vertex angle, and the perpendicular bisector of the base all at once — four different definitions collapse onto the same segment. That's why any two of a leg, the base, an altitude, an angle, the area or the perimeter are enough to determine the whole triangle.
Isosceles triangles, case by case
Two equal sides, angles from the vertex-angle split
The altitude from the apex cuts the vertex angle exactly in half and the base exactly in half. That produces two congruent right triangles with hypotenuse a, adjacent leg hb and opposite leg b/2. Every formula on this page comes from that one right triangle.
Area from a leg and the base
Once you know a and b, the height to the base is fixed by Pythagoras on the half-triangle. Multiplying half the base by that height gives the area — no trigonometry needed.
Both altitudes — hb (to base) and ha (to a leg)
hb is the axis of symmetry and the height most people mean. ha is the perpendicular from a base vertex to the opposite leg — smaller than hb whenever the triangle is 'tall', larger when it's squat. Both come out of the same area, computed two different ways.
The golden triangle (a special case)
A golden triangle is the isosceles triangle whose leg-to-base ratio equals φ = (1+√5)/2 ≈ 1.618. Its base angles are 72° and its vertex angle is 36°, meeting the constraint 2·72° + 36° = 180°. This is the triangle you see at the points of a regular pentagram and in the sequence of triangles that generates the golden spiral.
Notation used on this page
| Symbol | Meaning |
|---|---|
| a, c | The two equal legs (a = c by convention). |
| b | Base — the odd side, opposite the vertex. |
| A, C | Base angles (equal, opposite the equal legs). |
| B | Vertex angle (between the two legs, at the apex). |
| hb | Altitude from apex to base (the axis of symmetry). |
| ha, hc | Altitude from a base vertex to the opposite leg (equal to each other). |
| P, s | Perimeter and semiperimeter (s = P/2 = a + b/2). |
| K | Area. |
| r, R | Inradius and circumradius. |
Formulas at a glance
| Quantity | Formula | Where it comes from |
|---|---|---|
| Third side c | c = a | Two legs are equal by definition. |
| Base angles | A = C = (180° − B) / 2 | Angle sum minus the vertex angle, split evenly. |
| Base from leg + vertex | b = 2a · sin(B/2) | Half-triangle at the axis of symmetry. |
| Base angle from sides | A = arccos(b / 2a) | cos of base angle = adjacent (b/2) / hypotenuse (a). |
| Perimeter | P = 2a + b | Direct. |
| Semiperimeter | s = a + b/2 | P / 2. |
| Altitude to base | hb = √(4a² − b²) / 2 | Pythagoras on the half-triangle. |
| Altitude to leg | ha = hc = (b / 2a) · √(4a² − b²) | Two-area identity: ½·b·hb = ½·a·ha. |
| Area | K = (b / 4) · √(4a² − b²) = ½·a²·sin B | Base × height ÷ 2, or SAS area formula. |
| Inradius | r = K / s | Standard identity for every triangle. |
| Circumradius | R = a² / √(4a² − b²) | Law of sines: R = a / (2·sin A). |
Three ways to get the area
Depending on which measurements you already have, different area formulas save you an extra step:
The isosceles triangle theorem (and its converse)
The isosceles triangle theorem states: if two sides of a triangle are equal, the angles opposite those sides are equal. The converse also holds: if two angles of a triangle are equal, the sides opposite those angles are equal. The clean way to see both is to drop the altitude from the apex to the base — it splits the triangle into two right triangles that share the altitude and the two equal legs, so they're congruent by hypotenuse-leg. Corresponding angles match, and that gives A = C directly.
The golden triangle in a bit more depth
When the leg-to-base ratio is the golden ratio φ = (1 + √5)/2 ≈ 1.618, the triangle has base angles exactly 72° and vertex angle exactly 36°. Bisect one base angle and you cut off a smaller triangle that is itself a golden triangle — this self-similarity is what generates the logarithmic spiral that shows up in nautilus shells and sunflower heads. The five points of a regular pentagram are all golden triangles, and the "obtuse" golden triangle (also called the golden gnomon) — same φ ratio, but base angles 36° and vertex 108° — tiles the plane together with the acute golden triangle in Penrose tilings.
Features of this calculator
- Accepts any 2 of: leg a, base b, height to base hb, height to leg ha, vertex angle B, base angle A, area K, perimeter P.
- Degrees or radians toggle for every angle input and output.
- Live to-scale SVG showing labelled legs, base, vertex and base angles, tick marks on the equal legs, and the axis of symmetry hb.
- One-click presets for equilateral, golden triangle, 45-45-90 (isosceles right), and a roof-gable example.
- Auto-detects equilateral, right and golden cases and flags them in the result panel.
- Reports every side, angle, altitude, area, perimeter, semiperimeter, inradius r = K/s and circumradius R = a²/√(4a² − b²).
- Personalised step-by-step derivation using your actual numbers — every substitution shown.
Frequently asked questions
+What is an isosceles triangle?
An isosceles triangle has two sides of the same length (the legs). Because of that symmetry, the two angles opposite the equal legs — the base angles — are always equal to each other. The remaining angle, between the two equal legs, is called the vertex angle.
+How do I find the area of an isosceles triangle from the leg and base?
Drop a perpendicular from the apex to the base. It splits the triangle into two mirror-image right triangles with height hb = √(a² − (b/2)²) = √(4a² − b²)/2. Then the area is K = ½·b·hb = (b/4)·√(4a² − b²).
+How do I find the height of an isosceles triangle?
For the height from the apex to the base use hb = √(4a² − b²)/2. For the altitude drawn from a leg to the opposite vertex use ha = hc = (b/(2a))·√(4a² − b²). The base altitude hb is what people usually mean by 'the height'.
+How do I find a missing angle?
The three angles obey 2·A + B = 180°, where A is either base angle and B is the vertex angle. So if you know one angle you know the others: given B, each base angle is A = (180° − B)/2; given A, the vertex angle is B = 180° − 2A. If you know the sides, A = arccos(b / 2a).
+What is a golden triangle?
A golden triangle is an isosceles triangle whose leg-to-base ratio equals the golden ratio φ ≈ 1.618. Its base angles are exactly 72° and its vertex angle is exactly 36°. It appears in the points of a regular pentagon and in the growth curve of the logarithmic (golden) spiral. Use the 'Golden triangle' preset to load it.
+How is this different from your Triangle and Right Triangle calculators?
This calculator is specialised for triangles with two equal sides — it exploits the symmetry and needs only 2 measurements to lock everything down. Use the general Triangle Calculator when your triangle is scalene (SSS/SAS/ASA/AAS/SSA). Use the Right Triangle Calculator when one angle is exactly 90°. A triangle can also be both isosceles and right (the 45-45-90 preset), in which case both this page and the Right Triangle page give the same answer.