Isosceles Triangle Calculator
Enter any two known values for an isosceles triangle to calculate the edge lengths, altitude, angles, area, perimeter, inradius, and circumradius.
Solution:
What is an Isosceles Triangle?
An isosceles triangle is a triangle that has two edges, or legs, of the same length. The third edge is called the base.
The two angles adjacent to the base are called the base angles, while the angle opposite the base is called the vertex angle.
Because the legs are of equal length, the base angles are also identical.
Types of Isosceles Triangles
There are four types of isosceles triangles: acute, obtuse, equilateral, and right.
An acute isosceles triangle is a triangle with a vertex angle less than 90°, but not equal to 60°.
An obtuse isosceles triangle is a triangle with a vertex angle greater than 90°.
An equilateral isosceles triangle is a triangle with a vertex angle equal to 60°. An equilateral triangle is a special case where all the angles are equal to 60° and all three sides are equal in length. Try our equilateral triangle calculator.
A right isosceles triangle is a triangle with a vertex angle equal to 90°, and base angles equal to 45°. We have a special right triangle calculator to calculate this type of triangle.
How to Calculate Edge Lengths of an Isosceles Triangle
Given the height, or altitude, of an isosceles triangle and the length of one of the legs or the base, it’s possible to calculate the length of the other sides.
Solve the Base Length
Use the following formula to solve the length of the base edge:
b = 2a² – h²
The base length b is equal to 2 times the square root of leg a squared minus the height h squared.
Solve the Leg Length
Use the following formula to solve the length of the legs:
a = h² + (b ÷ 2)²
The leg length a is equal to the square root the height h squared plus the base b divided by 2, squared.
How to Calculate the Angles of an Isosceles Triangle
Given any angle in an isosceles triangle it is possible to solve the other angles.
Solve the Base Angle
Use the following formula to solve the base angle:
α = 180° – β2
The base angle α is equal to 180° minus vertex angle β, divided by 2.
Solve the Vertex Angle
Use the following formula to solve the vertex angle:
β = 180° – 2α
The vertex angle β is equal to 180° minus 2 times the base angle α.
How to Calculate Area and Perimeter
Given the sides of an isosceles triangle it is possible to solve the perimeter and area using a few simple formulas.
Solve Perimeter
Solve the perimeter of an isosceles triangle using the following formula:
p = 2a + b
Thus, the perimeter p is equal to 2 times leg a plus base b.
Solve Semiperimeter
Given the perimeter you can solve the semiperimeter. The semiperimeter s is equal to half the perimeter.
s = p2
Solve Area
To solve for area, use Heron’s formula:
T = s(s – a)(s – a)(s – b)
Heron’s formula states that the area T is equal to the square root of the semiperimeter s times semiperimeter s minus leg a times semiperimeter s minus a times semiperimeter s minus base b.
