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Equilateral Triangle Calculator
Input one element of an equilateral triangle, and the calculator will find the five unknown elements. The calculator will provide a step-by-step explanation on how to calculate missing elements.
Find the area $ A $ of an equilateral triangle if side $a = 12$.
solution
$$ A = 36 \sqrt{ 3 } $$explanation
To find area $ A $ use formula:
$$ A = \dfrac{ \sqrt{ 3 } \cdot a^2 }{ 4 } $$After substituting $ a = 12 $ we have:
$$ A = \dfrac{ \sqrt{ 3 } \cdot 12^2 }{ 4 } $$ $$ A = \dfrac{ \sqrt{ 3 } \cdot 144 }{ 4 }$$ $$ A = \dfrac{ 144 \sqrt{ 3 } }{ 4 }$$ $$ A = 36 \sqrt{ 3 } $$
| $$ A = \frac{3 \, a^2 \sqrt{3}}{4} $$ |
| area |
| $$ h = \frac{a \sqrt{3}}{2} $$ |
| height |
| $$ r = \frac{a \sqrt{3}}{6} $$ |
| incircle radius |
| $$ R = \frac{a \sqrt{3}}{3} $$ |
| circumcircle radius |
Equilateral triangle calculations
This calculator uses the following formulas to find the missing values of a triangle.
| Perimeter: | $$ P = 3 \cdot a $$ |  |
| Area: | $$ A = \frac{a^2 \sqrt{3}}{4} $$ | |
| Height: | $$ h = \frac{a \sqrt{3}}{2} $$ | |
| Circumcircle radius: | $$ R = \frac{a \sqrt{3}}{3} $$ | |
| Incircle radius: | $$ r = \frac{a \sqrt{3}}{6} $$ |
Example 01 :
What is the area of an equilateral triangle whose side is $ 12 cm $.
Solution:
In this example we have $ a = 12 $.
To find the area we will use formula $A = \dfrac{a^2 \sqrt{3}}{4} $
$$ \begin{aligned} A & = \frac{a^2 \sqrt{3}}{4} \\[ 1 em] A & = \frac{12^2 \sqrt{3}}{4} \\[ 1 em] A & = \frac{144 \sqrt{3}}{4} \\[ 1 em] A & = 36 \sqrt{3} \end{aligned} $$Example 02 :
What is the side of an equilateral triangle whose height is 15 cm?
Solution:
In this example we have $ h = 15 $.
To find height we will use formula $h = \dfrac{a \sqrt{3}}{2} $
$$ \begin{aligned} h & = \frac{a \sqrt{3}}{2} \\[ 1 em] 15 & = \frac{a \sqrt{3}}{2} \\[ 1 em] a \sqrt{3} & = 15 \cdot 2 \\[ 1 em] a \sqrt{3} & = 30 \\[1 em] a & = \frac{30}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \\[1 em] a & = \frac{30 \sqrt{3}}{3} \\[ 1 em] a & = 10 \sqrt{3} \approx 17.3 \end{aligned} $$