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Equilateral Triangle Calculator

Equilateral Triangle Calculator

Input one value for 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 side $ a $ of an equilateral triangle if altitude $h = 15$.

solution

$$ a = 10 \sqrt{ 3 } $$

explanation

To find side $ a $ use formula:

$$ h = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$

After substituting $ h = 15 $ we have:

$$ 15 = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$ $$ \sqrt{ 3 } \cdot a = 15 \cdot 2 $$ $$ \sqrt{ 3 } \cdot a = 30 $$ $$ a = \dfrac{ 30 }{ \sqrt{ 3 } } $$ $$ a = 10 \sqrt{ 3 } $$

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Input the side, perimeter, area, circumcircle radius or altitude of an equilateral triangle, then choose a missing value.

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 = $

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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.

Find more worked examples in popular problems.

TUTORIAL

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} $

$$ \begin{aligned} A & = \frac{a^2 \sqrt{3}}{4} \\[ 1 em] A & = \frac{12^2 \sqrt{3}}{4} \\[ 1 em] A & = \frac{144 \sqrt{3}}{4} \\[ 1 em] A & = 36 \sqrt{3} \end{aligned} $$

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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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