Complete isosceles triangle solver. Enter base + leg, base + height, or leg + apex angle to compute all sides, angles, area, perimeter, altitudes, circumradius, and inradius.
An isosceles triangle has two sides of equal length, called the legs, and a third side called the base. The two angles opposite the equal sides (base angles) are always equal. Isosceles triangles appear everywhere — from the gable of a roof to the cross-section of an A-frame cabin, from the shape of yield signs to the geometry of suspension bridges.
Computing the full set of properties requires only two independent measurements. Given the base b and leg a, the height h = √(a² − (b/2)²), the base angles equal arccos(b / (2a)), and the apex angle is 180° minus twice the base angle. Given the base and height, the leg is √((b/2)² + h²). Given a leg and the apex angle α, the base = 2a·sin(α/2).
Once all three sides and all three angles are known, derived quantities follow: area = ½bh, perimeter = 2a + b, circumradius R = (a²b) / (4·Area), inradius r = Area / s where s is the semi-perimeter, the altitude from a base vertex to the opposite leg, and the medians. Structurally, the altitude from the apex to the base is also the perpendicular bisector and the median — a key symmetry of isosceles triangles.
This calculator supports three input modes (base + leg, base + height, leg + apex angle), a unit selector, presets for common isosceles shapes, and a summary table of all properties.
Isosceles Triangle — Sides, Angles & Area problems often require several dependent steps, and a small arithmetic slip can propagate through every derived value. This calculator is tailored to that workflow: you enter base (b), leg (a), height (h), and it returns area, perimeter, height, apex angle in one consistent pass. It is useful for homework checks, worksheet generation, tutoring walkthroughs, and fast field/design estimates where you need reliable geometry results without rebuilding the full derivation each time.
Height from base: h = √(a² − (b/2)²) Base angles: β = arccos(b / (2a)) Apex angle: α = 180° − 2β Area: A = ½ × b × h Perimeter: P = 2a + b Semi-perimeter: s = (2a + b) / 2 Circumradius: R = (a² × b) / (4A) Inradius: r = A / s
Result: Height = 12, Area = 60, Perimeter = 36, Base angle ≈ 67.38°, Apex angle ≈ 45.24°
With base 10 and leg 13: height = √(169 − 25) = √144 = 12. Area = ½ × 10 × 12 = 60. Perimeter = 26 + 10 = 36. Base angle = arccos(5/13) ≈ 67.38°. Apex = 180 − 2(67.38) ≈ 45.24°.
This isosceles triangle — sides, angles & area tool links the entered values (base (b), leg (a), height (h), apex angle) to the target geometry relationships used in class and practice problems. Instead of solving each intermediate step manually, you can validate setup and arithmetic quickly while still tracing which measurements drive the final result.
Formula focus: the calculator formula
Isosceles Triangle — Sides, Angles & Area shows up in school geometry, technical drafting, construction layout checks, and early engineering design estimates. When values are changed repeatedly, the calculator helps you compare scenarios quickly and see how sensitive the shape is to each dimension.
Start with the primary outputs (area, perimeter, height, apex angle) and then use the remaining cards/tables to confirm consistency with your diagram. Keep units consistent across inputs, and round only at the end if your assignment or project specifies a fixed precision.
Having exactly two sides of equal length. The angles opposite those equal sides are also equal.
If you know the base b and leg a, the height h = √(a² − (b/2)²). The altitude drops from the apex perpendicular to the base.
Yes — when the apex angle is 90°. The two base angles are each 45°, giving a 45-45-90 triangle.
An equilateral triangle has all three sides equal (and all angles 60°). An isosceles has exactly two equal sides.
R = (a² × b) / (4 × Area), where a is the leg and b is the base. Use this as a practical reminder before finalizing the result.
The inradius is the radius of the inscribed circle: r = Area / semi-perimeter. Keep this note short and outcome-focused for reuse.