Equilateral Triangles - Calculator

This calculator can be used to calculate the altitude, area, perimeter, inradius and circumradius of an equilateral triangle.

Formulas

Altitude	ha = 3 ∙ a 2
Area	A = a ∙ ha 2 = 3 ∙ a² 4
Perimeter	P = 3 ∙ a
Inradius	r = a 2 ∙ 3
Circumradius	R = a 3

Side lengths and angles

An equilateral triangle, as the name implies, is a triangle where all 3 sides have the same length. Also, all 3 interior angles are 60° angles.

Altitudes

In an equilateral triangle, all 3 altitudes have the same length. Therefore, h_a = h_b = h_c is valid. Often the length of the 3 altitudes is called h and the letter to which side the altitude belongs is omitted.

The altitude h is calculated as follows:

Example:

The sides of an equilateral triangle are each 4 cm long. The altitude is to be calculated.

A	=	3∙a 2
	=	3∙4 cm 2
	=	3∙2 cm
	≈	3.46410 cm

Area

The area of a triangle is calculated by selecting one side as the base side, multiplying the length of the base side by the corresponding altitude, and dividing the result by 2.

If you choose the side with side length a as the base side and h_a is the corresponding altitude, then you calculate the area as follows:

If the side length is known, you can calculate the area directly without calculating the altitude first. To do this, insert the formula for the altitude into the equation. Then you get:

This can be further simplified:

Example:

All 3 sides of a triangle have a side length of 4 cm each. We are looking for the area.

A	=	3∙a² 4
	=	3∙(4 cm)² 4
	=	3∙16 cm² 4
	=	3∙4 cm²
	≈	6.92820 cm²

Alternatively, the area could also be calculated using the altitude of the triangle. This was calculated in the example from the chapter Altitudes and amounts to approximately 3.46410 cm.

A	=	a∙ha 2
	≈	4 cm∙3.46410 cm 2
	=	2 cm∙3.46410 cm
	=	6.92820 cm²

Perimeter

To calculate the perimeter of a triangle, the sum of the lengths of the 3 sides must be calculated. Since all 3 sides of an equilateral triangle have the same length, one calculates for a given side length a:

Inradius

An incircle of a triangle is the circle that lies inside the triangle and touches each side of the triangle at exactly one point. In an equilateral triangle, these points of contact are exactly in the middle of the sides.

In a triangle, the center of the incircle is where the 3 angle bisectors intersect. In an equilateral triangle, the 3 altitudes also intersect at this point.

The inradius can be calculated with the following formula:

Circumcircle

A circumcircle of a triangle is a circle whose circular line touches each vertex of the triangle.

The center of the circumcircle of a triangle is where the 3 perpendicular bisectors of the 3 sides intersect.

In an equilateral triangle, just as with the incircle, this point is where the 3 altitudes intersect.

The radius of the circumcircle can be calculated with the following formula: