# The area of a regular hexagon is 1500 square centimeters. What is its perimeter? Please show working.

Nov 24, 2015

The perimeter is approximately $144.24 c m$.

#### Explanation:

A regular hexagon consists of 6 congruent equilateral triangles, so its area can be calculated as:

$A = 6 \cdot \frac{{a}^{2} \sqrt{3}}{4} = 3 \cdot \frac{{a}^{2} \sqrt{3}}{2}$.

The area is given, so we can solve an equation:

$3 \cdot \frac{{a}^{2} \sqrt{3}}{2} = 1500$

to find the length of the hexagon's side

$3 \cdot \frac{{a}^{2} \sqrt{3}}{2} = 1500$

Multiplying by $2$

$3 \cdot \left({a}^{2} \cdot \sqrt{3}\right) = 3000$

Dividing by $3$

${a}^{2} \cdot \sqrt{3} = 1000$

For further calculations I take approximate value of $\sqrt{3}$

$\sqrt{3} \approx 1.73$

So the equality becomes:

$1.73 \cdot {a}^{2} \approx 1000$

${a}^{2} \approx 578.03$

$a \approx 24.04$

Now we can calculate the perimeter:

$P \approx 6 \cdot 24.04$

$P \approx 144.24$

Nov 24, 2015

$\text{perimeter"=144.17"cm}$

#### Explanation:

The hexagon can be split into 6 equilateral triangle.

Each triangle has area of frac{1500"cm"^2}{6}=250"cm"^2

If the length of each triangle is $l$, then the perimeter of the hexagon is simply $6 l$.

Looking at 1 triangle, the area is given by half x base x height.

The base is $l$. The height is found by cutting the triangle into half and applying Pythagoras theorem.

${h}^{2} + {\left(\frac{l}{2}\right)}^{2} = {l}^{2}$

$h = \frac{\sqrt{3}}{2} l$

$\text{Area} = \frac{1}{2} \cdot l \cdot h$

$= \frac{1}{2} \cdot l \cdot \frac{\sqrt{3}}{2} l$

$= \frac{\sqrt{3}}{4} {l}^{2}$

$= 250 {\text{cm}}^{2}$

$l = 24.028 \text{cm}$

$\text{perimeter"=6l=144.17"cm}$