Assume each side of the hexagon is worth 10 units. Calculate the apothem, using the fact that it is part of a special triangle to get the answer exactly. Then use the exact value of the apothem to calculate the area of the hexagon.
the apothem is the altitude of an equilateral triangle of side 10: 5√3
area = 1/2 apothem * perimeter = 1/2 * 5√3 * 10 = 25√3
5√3
To calculate the apothem of a regular hexagon using the fact that it is part of a special triangle, we need to consider a few properties of the triangle.
In a regular hexagon, each interior angle is 120 degrees. Bisecting one of these angles creates a right triangle within the hexagon, with the apothem as the hypotenuse. The other two sides of the triangle are half the length of the hexagon's side, because they coincide with the radii of the inscribed circle.
So, in this case, each side of the hexagon is worth 10 units, which means each side of the special triangle formed inside the hexagon is 5 units.
Now, using the Pythagorean theorem, we can find the length of the apothem. The formula is: apothem^2 = leg1^2 + leg2^2
In the special triangle, leg1 = 5 units and leg2 is the apothem we're trying to find.
So, apothem^2 = 5^2 + leg2^2
Simplifying the equation, we get: apothem^2 = 25 + leg2^2
Since the apothem is a part of the special triangle, we know that isosceles triangle with an angle of 30 degrees has two sides that are congruent. So, if we consider one of those congruent sides as leg2, we know both legs of the triangle have a length of 5 units.
Now, using the Pythagorean theorem again, we can calculate the length of the apothem.
So, apothem^2 = 25 + 5^2
Simplifying further, we get: apothem^2 = 25 + 25
Finally, apothem^2 = 50
Taking the square root of both sides, we find the apothem is approximately equal to 7.07 units.
Now, to calculate the area of the hexagon, we can use the formula: Area = 1/2 * apothem * perimeter
Since each side of the hexagon is 10 units, the perimeter is 6 * 10 = 60 units.
Using the exact value of the apothem we found (approximately 7.07 units), the area of the hexagon is: Area = 1/2 * 7.07 units * 60 units
Simplifying the calculation, we find the area of the hexagon is: Area = 212.1 square units.