Equilateral Triangle Calculator
Input one element of an equilateral triangle, and the calculator will find the five unknown elements. The calculator provides a step-by-step explanation on how to find missing elements.
Find the side $ a $ of an equilateral triangle if altitude $h = 15\, \text{cm}$.
solution
$$ a = 10 \sqrt{ 3 }\, \text{cm} $$explanation
To find side $ a $ use formula:
$$ h = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$After substituting $h = 15\, \text{cm}$ we have:
$$ 15\, \text{cm} = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$ $$ \sqrt{ 3 } \cdot a = 15\, \text{cm} \cdot 2 $$ $$ \sqrt{ 3 } \cdot a = 30\, \text{cm} $$ $$ a = \dfrac{ 30\, \text{cm} }{ \sqrt{ 3 } } $$ $$ a = 10 \sqrt{ 3 }\, \text{cm} $$Tutorial
What is an equilateral triangle?
An equilateral triangle is a triangle in which all sides are equal. Some main properties are:
1. All angles of an equilateral triangle are equal, and they measure 60°,
2. This is a regular polygon with three sides,
3. For a given side, the altitude, angle bisector, and median are all the same.
What are the most important formulas?
1. Height of Equilateral Triangle
The height of an equilateral triangle can be derived using pythagorean theorem and it is equal
h=√3·a/2
2. Area
For each triangle the area is calculated using formula A=a·h/2. When we substitute above formula we obtain formula for calculating equilateral trinagle area.
A=a2·√3/4
3. Incircle radius
A circle inscribed in an equilateral triangle has a radius equal to one-third of its height.
r=√3·a/6
4. Perimeter
TThe perimeter of each triangle is the sum of the lengths of all sides, so for the equilateral triangle we have:
P = 3·a
Worked examples
Example 01:
What is the area of an equilateral triangle whose side is a = 12 cm.
To find the area we will use formula A=a2·√3/4.
A = a2 √3/4
A = 122 √3/4
A = 1442 √3/4
A = 36 √3
Example 02:
What is the side of an equilateral triangle whose height is h = 15 cm?
To find height we will use formula h = a·√3/2