Isosceles Triangle Calculator

Welcome to the Isosceles Triangle Calculator – this tool is designed to make calculations of sides, angles, height, perimeter, and area a breeze.

An isosceles triangle is a unique geometric figure characterized by having two sides of equal length, forming two equal angles. The third side is of a different length, creating a sense of symmetry. If the third side is also of the same length, it becomes an equilateral triangle.Understanding the properties of isosceles triangles opens a door to a captivating realm of geometry.

Let’s unravel the magic of isosceles triangles through the Isosceles Triangle Calculator, offering step-by-step instructions for calculating various parameters.

Denoted as the waist side (a), the bottom side (b), and the other side(c).

In an isosceles triangle, the other side (c) is equal to the waist side (a). Therefore

c = a

Determine the two equal angles (θ) using trigonometric ratios. The height on the bottom side is also the center line on the bottom edge. Calculate for equal angles using arc cosine:

θ = arccos(b/2a)

Find the perimeter (P) by summing the three sides:

P = a + b + c

which simplifies to

P = a + b + a = 2a + b

Utilize Heron’s formula for the area (A):

A= √(s * (s – a) * (s – b) * (s – c))

Where, s is the semi-perimeter. In this case

s = (2a+b)/2 = a+ b/2

The height (h) of the isosceles triangle can be found using

h = √(a² – (b²/4))

Example, The waist length of an isosceles triangle is a = 4units and the base length is b = 6 units.

The equal angles are

θ = arccos(b/2a) = arccos(6 / 2 * 4) = arccos(3 / 4) = 41.4 degrees

The vertex angle is

180 – 2θ = 180 – 2 * 41.4 = 97.2 degrees

The perimeter is

P = 2a + b = 2 * 4 + 6 = 14 units

The semi-perimeter is

s = P/2 = 14 /2 = 7 units

The area is

A = √(s * (s – a) * (s – b) * (s – c))

A = √(7 * (7 – 4) * (7 – 6) * (7 – 4))

A = √(7 * 3 * 1 * 3)

A = √42

A = 7.937 square units

The height is

h = √(a² – (b²/4)) = √(4² – (6²/4)) = 2.65 units

Let’s work through the scenario where you have one angle and one side in an isosceles triangle. First, we need to determine whether the given angle is a base angle or a vertex angle. Then, we can use the fact that the sum of the interior angles of a triangle is equal to 180 degrees to calculate the three angles of the isosceles triangle.

If the given angle (θ) is one of the base angles, then the other base angle is also θ since an isosceles triangle has two equal angles. The vertex angle is

180 – 2θ

If the given angle (θ) is the vertex angle (angle between the two equal sides), then the base angles are also:

(180 – θ)/2

Now you know the three angles and one side of an isosceles triangle, and you want to calculate the other two sides (b and c) using the sine theorem.

B = a*sin(b)/sin(a)

 

C = a *sin(c)/sin(a)

Once you know the three angles and three sides of an isosceles triangle, you can easily calculate the perimeter, area, and height of the isosceles triangle. See method one for details.