problem
Find 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:
$$ 15 = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$ $$ \sqrt{ 3 } \cdot a = 15 \cdot 2 $$ $$ \sqrt{ 3 } \cdot a = 30 $$ $$ a = \dfrac{ 30 }{ \sqrt{ 3 } } $$ $$ a = 10 \sqrt{ 3 } $$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: | $$ r = \frac{a \sqrt{3}}{6} $$ |
What is the area of an equilateral triangle whose side is $ 12 cm $.
In this example we have $ a = 12 $.
To find the area we will use formula $A = \dfrac{a^2 \sqrt{3}}{4} $
$$ \begin{aligned} A & = \frac{a^2 \sqrt{3}}{4} \\[ 1 em] A & = \frac{12^2 \sqrt{3}}{4} \\[ 1 em] A & = \frac{144 \sqrt{3}}{4} \\[ 1 em] A & = 36 \sqrt{3} \end{aligned} $$What is the side of an equilateral triangle whose height is 15 cm?
In this example we have $ h = 15 $.
To find height we will use formula $h = \dfrac{a \sqrt{3}}{2} $
$$ \begin{aligned} h & = \frac{a \sqrt{3}}{2} \\[ 1 em] 15 & = \frac{a \sqrt{3}}{2} \\[ 1 em] a \sqrt{3} & = 15 \cdot 2 \\[ 1 em] a \sqrt{3} & = 30 \\[1 em] a & = \frac{30}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \\[1 em] a & = \frac{30 \sqrt{3}}{3} \\[ 1 em] a & = 10 \sqrt{3} \approx 17.3 \end{aligned} $$Please tell me how can I make this better.