Isosceles Triangle Angles
Calculate the base angles and vertex angle of any isosceles triangle from base and equal side lengths. Free online geometry solver with arccos formula and instant degree results.
About This Calculator
An isosceles triangle is a triangle with two sides of equal length, called the legs, and a third side called the base. Because the legs are equal, the two base angles (angles adjacent to the base) are also equal. The remaining angle at the top, formed by the two equal sides, is called the vertex angle. This calculator computes both the base angles and the vertex angle instantly from just the base and equal side lengths.
The formula used is derived from basic trigonometry. By splitting the isosceles triangle down its axis of symmetry (from the vertex angle to the midpoint of the base), we get two congruent right triangles. In each right triangle, the adjacent side is half the base (b/2), and the hypotenuse is the equal side length (s). The base angle alpha = arccos(b / (2s)) in radians, then converted to degrees by multiplying by 180/pi. The vertex angle beta = 180 deg - 2alpha. For example, an isosceles triangle with base 10 and equal sides 13 has base angle ≈ 67.38 deg and vertex angle ≈ 45.24 deg.
Isosceles triangles appear throughout geometry, architecture (roof trusses, gable ends, bridge supports), engineering (structural bracing, truss designs), and design (logo shapes, packaging). This tool is ideal for students learning trigonometry, teachers preparing geometry problems, architects checking roof pitches, carpenters calculating rafter angles, and DIY enthusiasts needing quick angle measurements without manual trigonometric calculations.
Regional Notes: Triangle geometry is universal and does not vary by region. The calculator works with any unit system -- enter the base and equal sides in the same unit (cm, inches, feet, meters, yards, etc.) and the angles are reported in degrees. No currency or region-specific settings are needed since angles are dimensionless quantities derived from side-length ratios.
Key facts about isosceles triangle angles: (1) Base angles are always equal. (2) If the base equals the legs, all three angles are 60 deg (equilateral triangle, a special case of isosceles). (3) If the vertex angle is 90 deg, the triangle is a right isosceles triangle with 45 deg base angles. (4) The angles depend only on the ratio of base to side length, not on the absolute size of the triangle.
Frequently Asked Questions
What are the angles of an isosceles triangle?
An isosceles triangle has three angles: two equal base angles and one vertex angle. The base angles are the angles adjacent to the base side, and they are always equal. The vertex angle is the angle formed by the two equal sides (legs) at the top. The sum of all three angles is always 180 degrees.
How do I find the base angles of an isosceles triangle?
The base angles of an isosceles triangle can be found using trigonometry. If you know the base length b and equal side length s, the base angle alpha = arccos(b / (2s)). For example, a triangle with base 10 and equal sides 13 has base angle alpha = arccos(10/26) ≈ 67.38 deg. Both base angles are equal.
How do I find the vertex angle of an isosceles triangle?
The vertex angle of an isosceles triangle is found by subtracting the two base angles from 180 degrees. Since both base angles are equal, the formula is vertex angle = 180 deg - 2 x base angle. Using the arccos formula for the base angle, vertex angle = 180 deg - 2 x arccos(b / (2s)).
Can an isosceles triangle have a right angle?
Yes, an isosceles triangle can have a 90-degree angle. This angle must be the vertex angle, forming an isosceles right triangle (also called a 45-45-90 triangle). The two base angles are then 45 degrees each. A right isosceles triangle has legs of equal length and the hypotenuse as the base.
What is the condition for a valid isosceles triangle?
For an isosceles triangle with equal sides s and base b, the triangle inequality requires that 2s > b -- the sum of the two equal sides must exceed the base. Equivalently, each equal side must be longer than half the base (s > b/2). If the equal sides are equal to or shorter than half the base, the triangle cannot exist and the angles are undefined.
What is the sum of angles in an isosceles triangle?
The sum of all three interior angles in any isosceles triangle is 180 degrees. This follows the general triangle angle sum theorem. In an isosceles triangle, this is expressed as 2 x base angle + vertex angle = 180 deg.
Is this calculator free to use?
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