Calculate the apex angle and base angles of an isosceles triangle from the equal sides and base. Verifies the angle sum equals 180° and shows area, perimeter, height.
An isosceles triangle has two equal sides (a) and a base (b). Because the two equal sides mirror each other, the angles opposite them — the base angles — are also equal. The remaining angle at the top, between the two equal sides, is called the apex (or vertex) angle.
Given the side lengths, you can find the apex angle using the inverse sine function: α = 2·arcsin(b / (2a)). The two base angles follow immediately: β = (180° − α) / 2. These three angles always sum to exactly 180°.
This calculator takes the equal side a and the base b, computes both angles, and verifies the 180° sum. It also derives secondary properties such as the height, area, perimeter, inradius, and circumradius of the triangle. The angle output can be displayed in degrees or radians.
A visual bar chart shows how the three angles partition the full 180°, making it easy to see whether the triangle is acute, right, or obtuse. A reference table of notable isosceles triangles — including the equilateral triangle, the right isosceles triangle, the golden gnomon, and the golden triangle — lets you compare your results with well-known shapes.
This tool is useful for students studying geometry, engineers analyzing symmetrical structures, and designers working with triangular motifs.
Isosceles Triangle Angles 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 equal side a (value), base b (value), unit, and it returns apex angle (vertex), base angle, angle sum check, height 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.
Apex angle α = 2·arcsin(b / (2a)). Base angle β = (180° − α) / 2. Height h = √(a² − (b/2)²). Area = ½·b·h.
Result: Apex angle ≈ 73.74°, Base angle ≈ 53.13°
α = 2·arcsin(3/5) = 2·arcsin(0.6) ≈ 2 × 36.87° = 73.74°. β = (180 − 73.74)/2 ≈ 53.13°. Sum = 73.74 + 53.13 + 53.13 = 180°.
This isosceles triangle angles tool links the entered values (equal side a (value), base b (value), unit, angle 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 Angles 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 (apex angle (vertex), base angle, angle sum check, height) 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.
Use α = 2·arcsin(b / (2a)), where a is the equal side and b is the base. Use this as a practical reminder before finalizing the result.
Because the two equal sides create mirror symmetry about the altitude from the apex, making the angles at the base congruent. Keep this note short and outcome-focused for reuse.
Yes. When b > a√2, the apex angle exceeds 90° and the triangle is obtuse.
The triangle degenerates — the apex angle reaches 180° and the points become collinear. Apply this check where your workflow is most sensitive.
It should always be exactly 180° mathematically. Any tiny deviation (like 179.9999°) is floating-point rounding, not a real error.
A special isosceles triangle with a 36° apex angle whose sides are in the golden ratio (φ ≈ 1.618). It appears in regular pentagons and Penrose tilings.