Isosceles and equilateral triangles are special because their equal sides create equal angles, which unlocks shortcuts for finding missing measurements. You can check your work with the free tools on Calc Solver.
What Makes a Triangle Isosceles
An isosceles triangle has exactly two equal sides, and the angles opposite those sides, called base angles, are equal. The third side is the base, and the angle opposite it is the vertex angle. Spotting the equal sides immediately tells you which angles match.
What Makes a Triangle Equilateral
An equilateral triangle has all three sides equal, so all three angles are equal too. Since the angles of any triangle sum to 180 degrees, each angle of an equilateral triangle is exactly 60 degrees. One known side or one known angle determines the whole triangle.
Finding Missing Angles
Use the angle sum with the equal-angle facts. If an isosceles triangle has a vertex angle of 50 degrees, the two base angles share the remaining 130 degrees, so each is 65 degrees. If an isosceles triangle has a base angle of 70 degrees, the other base angle is also 70 degrees, and the vertex angle is 180 minus 140, which is 40 degrees.
Finding Missing Sides
In an isosceles triangle, the sides opposite equal angles are equal, so one known leg gives you the other. In an equilateral triangle, one known side gives you all three, and the perimeter is simply three times that side. The height of an equilateral triangle with side s is (sqrt(3)/2) times s, which splits it into two 30-60-90 right triangles.
Common Mistakes to Avoid
First, assuming a triangle is isosceles from its drawing alone: only side marks or given information prove it. Second, applying equilateral shortcuts to a merely isosceles triangle. Third, mixing up which angles are the base angles when the triangle is drawn rotated.
Practice Problems to Try
An isosceles triangle has a vertex angle of 100 degrees: find the base angles. An equilateral triangle has side 6: find its perimeter and each angle. An isosceles triangle has a base angle of 35 degrees: find the vertex angle.