To find the perimeter of a regular octagon, we need to know the length of one of its sides. We can use the fact that a regular octagon can be divided into eight congruent isosceles triangles, and the central angle of each triangle is 45 degrees.
Drawing a line from the center of the octagon to the midpoint of one of its sides will create a right triangle, where the hypotenuse is the radius of the octagon (which is 10m), and the two legs are half of the length of one of its sides.
Using the Pythagorean theorem, we can solve for the length of one of the sides of the octagon:
hypotenuse^2 = leg^2 + leg^2
10^2 = (side/2)^2 + (side/2)^2
100 = side^2/2
side = √(200) = 10√2 m
So the length of one side of the octagon is 10√2 m. Therefore, the perimeter of the octagon is:
perimeter = 8 x side = 8 x 10√2 = 80√2 m
To find the area of the octagon, we can use the formula:
area = (apothem x perimeter)/2
where the apothem is the distance from the center of the octagon to the midpoint of one of its sides. Using the same right triangle we used before, we can see that the apothem is equal to:
apothem = leg = 10√2/2 = 5√2 m
Therefore, the area of the octagon is:
area = (5√2 x 80√2)/2 = 200 x 2 = 400 m^2
So the perimeter of the octagon is 80√2 m and the area is 400 m^2.