Find the equal side (a) of an isosceles triangle given the base with height, area, perimeter, or apex angle. Shows all triangle properties.
An isosceles triangle has two sides of equal length, traditionally called side a, and a third side called the base (b). Given the base and one additional measurement — the height, the area, the perimeter, or the apex angle — you can determine the length of side a and every other property of the triangle.
When you know the base and height, the equal side is found directly with the Pythagorean theorem: a = √((b/2)² + h²). When the area is given instead, first recover the height as h = 2A/b, then apply the same formula. If the perimeter P is known, the equal side is simply a = (P − b)/2. And when the apex angle α is provided, trigonometry yields a = (b/2) / cos(α/2).
This calculator supports all four input modes and automatically computes the full set of triangle properties: side a, height, area, perimeter, apex angle, base angles, inradius, and circumradius. It includes eight quick-load presets, a comparison bar chart, a properties table, and a reference table of notable isosceles triangles so you can cross-check your results.
Whether you are solving a geometry homework problem, designing a gable roof, or laying out a triangular garden bed, this tool gives you every measurement you need in one place.
Isosceles Triangle Side a 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 (value), input mode, unit, and it returns equal side a, height, area, perimeter 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.
From base & height: a = √((b/2)² + h²). From base & area: h = 2A/b, then same formula. From base & perimeter: a = (P − b)/2. From base & apex angle: a = (b/2) / cos(α/2).
Result: Side a = 5 cm
a = √((6/2)² + 4²) = √(9 + 16) = √25 = 5 cm. Area = ½ × 6 × 4 = 12 cm². Perimeter = 2(5) + 6 = 16 cm.
This isosceles triangle side a tool links the entered values (base b (value), input mode, unit) 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 Side a 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 (equal side a, height, area, perimeter) 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.
Side a refers to the two equal-length sides (legs). The third side is the base, b.
No — you need at least one additional measurement (height, area, perimeter, or an angle) to determine side a uniquely. Use this as a practical reminder before finalizing the result.
That is geometrically impossible. If h > a, the inputs are inconsistent (b/2 > a, meaning the base is too long).
Inradius r = Area / semi-perimeter. Circumradius R = (a²·b) / (4·Area). Both depend on a, b, and the derived height.
Yes. Enter an apex angle of 90° (or use base + height where h = b/2) and the calculator handles it correctly.
Yes. An obtuse isosceles triangle has an apex angle between 90° and 180° — the calculator supports this.