Isosceles Triangle Angles Calculator

Enter the leg length and base length of an isosceles triangle to find its vertex (apex) angle and its two equal base angles, plus the triangle's height and area.

Quick Facts

Base angle formula
cos(base angle) = (b/2) ÷ leg
Both base angles are equal since they sit opposite the two equal legs.
Vertex angle formula
vertex = 180° − 2 × base angle
Follows directly from the triangle angle sum (180°).
Validity condition
base < 2 × leg
The triangle inequality — otherwise the legs can't meet to close the triangle.

Your Results

Calculated
Vertex Angle (Apex)
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Angle between the two equal legs
Base Angles (each)
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The two equal angles at the base
Triangle Height
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Perpendicular distance from apex to base
Area
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½ × base × height

Ready

Enter the leg length and base length, then press Calculate.

Formula and Method for Isosceles Triangle Angles

An isosceles triangle has two equal sides, called legs, and a third side called the base. Because two sides are equal, the two angles opposite them — the base angles — are also equal. The angle between the two legs, opposite the base, is the vertex (apex) angle. Since every triangle's interior angles sum to 180°, knowing the leg length and base length is enough to solve for all three angles.

How the calculation works

Drop a perpendicular line from the apex straight down to the base. This altitude bisects the base into two equal halves (b/2) and splits the isosceles triangle into two congruent right triangles, each with the leg as the hypotenuse. In that right triangle, the base angle sits between the base and the leg, so cos(base angle) = (b/2) ÷ leg, giving base angle = arccos((b/2) ÷ leg). Once you have one base angle, the vertex angle follows from the angle sum: vertex angle = 180° − 2 × base angle. The same altitude also gives the triangle's height, h = √(leg² − (b/2)²), which the calculator uses to report area = ½ × base × height.

Common mistakes

  • Mixing up vertex vs. base angles: the vertex (apex) angle is between the two equal legs; the two base angles sit at the ends of the base and are always equal to each other.
  • Violating the triangle inequality: the base must be shorter than twice the leg length (base < 2 × leg). If base ≥ 2 × leg, the two legs cannot meet and no triangle exists.
  • Assuming a right isosceles triangle: not every isosceles triangle has a 90° angle — only enter a right-angle assumption if the problem actually states it.

Real-world applications

  • Roof trusses and gable ends are commonly isosceles, so the vertex and base angles determine rafter cuts and pitch.
  • A-frame structures, tents, and bridge trusses use isosceles triangles for symmetric load distribution.
  • Surveying and CAD/drafting use the base-angle formula to lay out symmetric wedge shapes precisely.
  • Geometry and trigonometry coursework use isosceles triangles to introduce the law of cosines and angle-sum reasoning.

Frequently Asked Questions

What is the formula for the base angles of an isosceles triangle?
Drop a perpendicular from the apex to the base; it bisects the base and forms a right triangle with the leg as hypotenuse. So cos(base angle) = (base/2) / leg, which gives base angle = arccos((base/2) / leg). Both base angles are equal because they are opposite the two equal legs.
How do I find the vertex (apex) angle?
Since a triangle's interior angles sum to 180°, the vertex angle equals 180° minus the two equal base angles: vertex angle = 180° − 2 × base angle. You can also get it directly from the law of cosines: vertex angle = arccos(1 − base² / (2 × leg²)).
What if I already know one angle instead of the side lengths?
If you know the vertex angle, each base angle is (180° − vertex angle) / 2. If you know one base angle, the vertex angle is 180° − 2 × base angle and the other base angle equals the first, since base angles are always equal in an isosceles triangle.
What is the maximum base length for a given leg length?
The base must be shorter than twice the leg length (base < 2 × leg) by the triangle inequality. As the base approaches 2 × leg, the vertex angle approaches 180° and the triangle flattens; there is no valid triangle at or beyond that limit.

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