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Equilateral Triangle Calculator

problem

Find 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:

$$ A = \dfrac{ \sqrt{ 3 } \cdot 12^2 }{ 4 } $$ $$ A = \dfrac{ \sqrt{ 3 } \cdot 144 }{ 4 }$$ $$ A = \dfrac{ 144 \sqrt{ 3 } }{ 4 }$$ $$ A = 36 \sqrt{ 3 } $$

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Input the side, perimeter, area, circumcircle radius or altitude of an equilateral triangle, then choose a missing value.
The calculator will display step-by-step explanation on how to find the missing value.
show help ↓↓ examples ↓↓ tutorial ↓↓
The missing value:
Provide any value for an equilateral triangle.
Calculator works with decimals, fractions and square roots (to input $ \color{blue}{\sqrt{2}} $ type $\color{blue}{\text{r2}} $)
side
$ a = $
 
height
$ h = $
perimeter
$ P = $
 
area
$ A = $
circumcircle
radius
$ R = $
 
incircle radius
$ r = $
 
working...
Equilateral triangle formulas
$$ A = \frac{3 \, a^2 \sqrt{3}}{4} $$
area
$$ h = \frac{a \sqrt{3}}{2} $$
height
$$ r = \frac{a \sqrt{3}}{6} $$
incircle radius
$$ R = \frac{a \sqrt{3}}{3} $$
circumcircle radius
EXAMPLES
example 1:ex 1:
What is the area of an equilateral triangle of perimeter $P = 6\sqrt{2}$.
example 2:ex 2:
What is the perimeter of an equilateral triangle if its height is $\dfrac{20}{3} cm^2$?
example 3:ex 3:
If base of an equilateral triangle 50 inches long, what is the triangle's height?
example 4:ex 4:
$\triangle ABC$ is an equilateral triangle with area A = 24. Find the perimeter.
TUTORIAL

Equilateral triangle calculations

This calculator uses the following formulas to find the missing values of a triangle.

Perimeter: $$ P = 3 \cdot a $$ equilateral triangle
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} $

$$ \begin{aligned} h & = \frac{a \sqrt{3}}{2} \\[ 1 em] 15 & = \frac{a \sqrt{3}}{2} \\[ 1 em] a \sqrt{3} & = 15 \cdot 2 \\[ 1 em] a \sqrt{3} & = 30 \\[1 em] a & = \frac{30}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \\[1 em] a & = \frac{30 \sqrt{3}}{3} \\[ 1 em] a & = 10 \sqrt{3} \approx 17.3 \end{aligned} $$
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