The area of a regular polygon is half the product of the apothem and the perimeter. We will first find the apothem and then the side length to obtain the perimeter. Finally, we will use this information to find the area.
Let's do it!
Apothem
By drawing the four radii, we can divide the square into four isosceles triangles. Since the triangles are congruent and a full turn measures 360^(∘), the central angles of the isosceles triangles formed by the radii measure 3604=90^(∘).
Now, let's consider just one of these isosceles triangles. We will also draw the apothem a of the hexagon, which is perpendicular to the side.
Remember that an apothem bisects the central angle and the side of the regular polygon. Therefore, we obtain a right triangle with an acute angle that measures 45^(∘).
By the Triangle Angle Sum Theorem we know that the sum of the three interior angles of a triangle add to 180^(∘). With this information we can find the measure of the unknown acute angle.
180- 90- 45=45^(∘)
The third angle of the right triangle measures 45^(∘). Therefore, we have a 45^(∘)-45^(∘)-90^(∘) triangles.
In this type of special triangle, the legs are congruent and length of the leg is sqrt(2)2 times the length of a hypotenuse.
Legs: a= sqrt(2)2 * 6
Let's simplify the right-hand side of this equation to obtain the apothem.
Consider the 45^(∘)-45^(∘)-90^(∘) triangle one more time. Note that in this kind of a triangle the legs are congruent.
As previously mentioned, the apothem bisects the side of the regular hexagon. Therefore, the length of the side of the given polygon is twice the length of the side of the above triangle.
Consequently, the side length of the square is 6sqrt(2)cm. Since this polygon has four congruent sides, to find its perimeter we will multiply the side length by 4.
Perimeter: 4* 6sqrt(2)=24sqrt(2)cm
Area
Now that we know that the apothem of the figure is 3sqrt(2)cm and that the perimeter is 24sqrt(2)cm. To find its area, we will substitute these values in the formula A= 12ap. Let's do it!
