Calculate all properties of an isosceles triangle including sides, angles, area, perimeter, and height. Step-by-step solutions for any combination of known values.
Calculate the properties of an isosceles triangle (two equal sides) with step-by-step solutions:
Provide at least 2 values. Common combinations: two equal sides and base, or base and height.
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An isosceles triangle is a triangle with at least two equal sides. The two equal sides are called legs, and the third side is called the base. Isosceles triangles have special properties that make calculations easier than general triangles.
Base Angle + Base Angle + Vertex Angle = 180°
Since base angles are equal: 2 × Base Angle + Vertex Angle = 180°
Our calculator can solve for any missing properties of an isosceles triangle when you provide at least 2 known values. The calculator automatically determines which method to use based on your inputs.
The following formulas are used to calculate properties of an isosceles triangle:
Where a is the equal side length and b is the base
Standard triangle area formula
Where a is the equal side and b is the base
Using cosine law in the right triangle formed by the height
Since all angles sum to 180°
Given: Equal sides = 10 units, Base = 12 units
Given: Equal side = 13 units, Height = 12 units
Given: Equal side = 8 units, Base = 6 units
Some isosceles triangles have additional special properties:
Isosceles triangles are commonly found in many practical applications:
Our calculator intelligently handles different input combinations:
When you know the two equal sides and the base, the calculator uses the Pythagorean theorem to find the height, then calculates all angles and properties.
When you know the base and height, the calculator finds the equal sides using the Pythagorean theorem, then calculates all angles.
When you know the base and one base angle, the calculator uses trigonometry to find the equal sides and height, then calculates the vertex angle.
When you know one equal side and the vertex angle, the calculator finds the base angles (which are equal), then calculates the base and height using trigonometry.
Explore these related geometry calculators to solve more triangle and geometric problems:
An isosceles triangle is a triangle with at least two equal sides. The two equal sides are called legs, and the third side is called the base. The base angles are equal, and the vertex angle is different.
The height can be found using the Pythagorean theorem: h = √(a² - (b/2)²), where a is the equal side length and b is the base. The height bisects the base and creates two congruent right triangles.
Yes, in an isosceles triangle, the base angles (the angles at the base) are always equal. This is a fundamental property of isosceles triangles.
An isosceles triangle has two equal sides, while an equilateral triangle has all three sides equal. An equilateral triangle is a special case of an isosceles triangle where all sides and angles are equal (60° each).
The area of an isosceles triangle is calculated using the standard triangle area formula: Area = (1/2) × base × height. You can also use Heron's formula if you know all three side lengths.
Yes, an isosceles right triangle (45-45-90 triangle) has two equal legs and a right angle at the vertex. The base angles are 45° each, and the hypotenuse is √2 times the length of each leg.
The vertex angle is the angle opposite the base, formed by the two equal sides. It can be calculated as: Vertex Angle = 180° - 2 × Base Angle, since all angles in a triangle sum to 180°.
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