Isosceles triangle from leg and apex angle calculator

It returns the base length, both equal base angles, the perpendicular height, and the enclosed area. The calculation splits the triangle into two congruent right triangles, making the half-angle relationships direct and easy to audit. Enter lengths in any consistent unit; the returned lengths use that unit and the area uses its square.

Turn the isosceles triangle into two right triangles

An isosceles triangle has two equal legs, and the apex angle lies between them. Draw the altitude from the apex to the base. In this particular triangle, that altitude is also a median and an angle bisector, so it divides the original shape into two congruent right triangles. Each right triangle has the supplied leg length as its hypotenuse, half of the apex angle at the top, half of the unknown base along the bottom, and the unknown height as the other perpendicular side. This construction is the key to the whole calculation. If the equal leg is L and the apex angle is A degrees, the half-angle is A/2. The half-base is L times sin(A/2), while the height is L times cos(A/2). Doubling the half-base gives the complete base. Because every step follows a right-triangle relation, the result does not depend on a diagram being drawn to scale, and no iterative approximation or ambiguous triangle case is involved.

How every returned measurement is calculated

The calculator first converts half of the apex angle from degrees to radians for the trigonometric functions. It computes the full base as 2L sin(A/2) and the height as L cos(A/2). The two base angles are equal because the legs are equal, and the interior angles of a triangle sum to 180 degrees. Therefore each base angle is (180 − A)/2. Finally, the area is one half times the complete base times the perpendicular height. This is equivalent to one half L squared times sin(A), but using the already calculated base and height keeps the returned values internally connected. Results are rounded only to a stable floating-point precision, not to a fixed number of decimal places. The leg, base, and height all share the unit you choose. If the leg is entered in centimeters, those three lengths are centimeters and the area is square centimeters. The angles are always returned in degrees.

Choose valid inputs and interpret extreme shapes

Use a finite leg length greater than zero and an apex angle strictly between 0 and 180 degrees. The endpoints are excluded because they collapse the triangle: at zero degrees the legs overlap, while at 180 degrees the height becomes zero and the sides form a straight segment. Angles very close to either endpoint are mathematically valid but describe very thin triangles. Near zero degrees, the base and area become small while the height approaches the leg length. Near 180 degrees, the base approaches twice the leg length while the height and area approach zero. These behaviors are useful checks when reviewing a result. The calculator does not attach or convert physical units, so all lengths must be interpreted consistently. Do not enter one leg in meters and expect a base in centimeters. For construction or manufacturing, retain enough digits for the required tolerance and apply project-specific rounding only after the geometric result has been calculated. The API price is $0.002 per calculation.