Find the area of the regular hexagon below by using the area formula for triangles.
To use the area formula for triangles, we need to divide the hexagon into six congruent triangles. Draw lines from the center of the hexagon to each vertex, dividing the hexagon into six congruent equilateral triangles.
Each of these triangles has a base of 10 and a height of $\frac{10\sqrt{3}}{2}=5\sqrt{3}$. Therefore, the area of each triangle is $\frac{1}{2}(10)(5\sqrt{3})=25\sqrt{3}$.
Since there are six congruent triangles in the hexagon, the total area of the hexagon is $6(25\sqrt{3})=\boxed{150\sqrt{3}}$.
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We can use the formula for the area of a regular hexagon, which is:
$A=\frac{3\sqrt{3}}{2}s^2$
where $s$ is the length of one side of the hexagon.
Since all sides of a regular hexagon are congruent, we can let $s=10$ (given in the diagram), and substitute into the formula:
$A=\frac{3\sqrt{3}}{2}(10)^2$
Simplifying:
$A=\frac{3\sqrt{3}}{2}(100)$
$A=150\sqrt{3}$
Therefore, the area of the regular hexagon is $\boxed{150\sqrt{3}}$.
To find the area of a regular hexagon using the area formula for triangles, follow these steps:
1. Start by dividing the regular hexagon into six congruent equilateral triangles. Since each angle in a regular hexagon is 120 degrees, each angle in an equilateral triangle is also 60 degrees.
2. Find the length of one side of the equilateral triangle. To do this, we need to use the formula for the length of a side of an equilateral triangle:
length of one side = (S/p) * 2sqrt(3)
where S is the length of the regular hexagon's side, and p is the perimeter of the hexagon.
3. Calculate the perimeter of the regular hexagon. Since all sides of a regular hexagon are equal, multiply the length of one side by 6:
perimeter = S * 6
4. Substitute the perimeter value back into the side length formula to find the length of one side:
length of one side = (S/ (S * 6)) * 2sqrt(3)
5. Simplify the expression:
length of one side = 2sqrt(3) / 6
6. Substitute the side length back into the equation for the area of an equilateral triangle:
Area of one triangle = (side length^2 * sqrt(3)) / 4
7. Substitute the side length value into the equation:
Area of one triangle = ( (2sqrt(3) / 6)^2 * sqrt(3)) / 4
8. Simplify the expression further:
Area of one triangle = (12 * sqrt(3)) / 72
9. Multiply the area of one triangle by 6 to find the total area of the regular hexagon:
Total area of the hexagon = 6 * (12 * sqrt(3)) / 72
10. Simplify the expression:
Total area of the hexagon = (72 * sqrt(3)) / 72
11. Cancel out the common factor of 72:
Total area of the hexagon = sqrt(3)
Therefore, the area of the regular hexagon is sqrt(3).