Start by finding the perimeter. Then, find the apothem. Finally, substitute the values of the perimeter and the apothem into the formula A= 12ap to find the area of the polygon.
We will first calculate the perimeter. Then, we will find the apothem.
Finally, since the area of a regular polygon is half the product of the apothem and the perimeter, we will use the obtained values to find the area. Let's do it!
Perimeter
Since we have a 10-sided regular polygon we can find the perimeter by multiplying 10 by the side length.
Perimeter= 10* 6=60cm
The perimeter of the given regular polygon is 60cm.
Apothem and Area
By drawing the 10 radii we can divide the polygon into 10 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 36010=36^(∘).
Now, let's consider just one of the isosceles triangles and the side length 6cm.
The apothem bisects the side of the polygon. Therefore, the shorter leg of the triangle is half the length of the side length. Let's consider the right triangle whose longer leg is the apothem of the polygon.
Remember, this a right triangle with a shorter leg that measures 3cm. To find the length of the side that is adjacent to the acute angle that measures 18^(∘), we will use the tangent ratio.
tan θ =Opposite/Adjacent
Let's substitute the corresponding values into this ratio and solve for a — the length of the longer leg of the triangle — and the shorter leg of the polygon.
We found that the apothem is 3tan 18^(∘)cm. We already found that the perimeter is about 60cm. To find the area, we will substitute these values in the formula A= 12ap.
