Problems from The regular polygon

Development:

1. We define a radius $r=10.$

2. The hexagon is formed by $6$ equilateral triangles. If we were to draw it, we will see that two of the sides of every equilateral triangle coincide with the radius of the circumference. Being an equilateral triangle, the third side, which coincides with the side of the hexagon, will be equal to the radius of the circumference. So: $$l=10$$

3. To find the area of the hexagon, we calculate first the area of one of the $6$ triangles that form it and multiply it by $6$.

The height $a$ of one of the $6$ triangles is:

$$l^2=a^2+({\displaystyle \frac{1}{2}}{)}^2$$ $$a^2=l^2(1-{\displaystyle \frac{1}{4}})$$ $$a={\displaystyle \frac{\sqrt{3}}{2}}\cdot l$$

The area of one of the $6$ triangles is:

$$A_{triangle}={\displaystyle \frac{l\cdot a}{2}}=l^2{\displaystyle \frac{\sqrt{3}}{4}}=25\sqrt{3}$$

And, multiplying by $6$:

$$A_{hexagon}=150\sqrt{3}$$

1. The area of the circumference is: $$A_{circ}=100\pi$$

And the percentage covered by the hexagon will be:

$$100\cdot {\displaystyle \frac{A_{hex}}{A_{circ}}}=100\cdot {\displaystyle \frac{150\sqrt{3}}{100\pi }}=82,7\mathrm{\%}$$

Solution:

1. $r=10$
2. $l=10$
3. $A_{hexagon}=150\sqrt{3}$
4. $82,7\mathrm{\%}$

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