find the ara of a regular octagon inscribed in a circle with a radius of 1 cm.

The octagon area can be broken into eight congruent isosceles triangles or sixteen congruent right triangles. The hypotenuse of each of the right trangles is r = 1 cm and the smallest angle is 360/16 = 22.5 degrees.

The area of each of the sixteen triangles is
A = (1/2)*r*cos 22.5*r sin 22.5
= (r^2/4)sin 45
Multiply this by 16 for the full area:
4 r^2*(sqrt 2)/2 = 2*sqrt 2* r^2 = 2.828 r^2
Note that this is less than pi*r^2, the circumscribed circle area, as it must be

If you know two sides of a triangle and the angle contained between those sides, then the area is

(1/2)(side1)(side2)sin(angle)

This is the case for one of the eight isosceles triangles
Each of those is (1/2)(1)(1)sin45º or .353553 or √2/4
so the octogon is 8*√2/4 or 2.828

To find the area of a regular octagon inscribed in a circle, we can use the following formula:

Area of octagon = 2 * (1 + √2) * (radius)^2

Given that the radius of the circle is 1 cm, we can substitute this value into the formula:

Area of octagon = 2 * (1 + √2) * (1)^2

Simplifying further:

Area of octagon = 2 * (1 + √2) * 1

Area of octagon = 2 + 2√2 square cm

To find the area of a regular octagon inscribed in a circle with a radius of 1 cm, we can use a simple formula.

Step 1: Determine the side length of the octagon.
In a regular octagon, all the sides are equal in length. We can find the side length by dividing the circumference of the circle by 8 (since an octagon has 8 sides).

Circumference = 2πr
Circumference = 2π(1 cm) = 2π cm

Side length = Circumference / 8 = (2π cm) / 8 = (π/4) cm

Step 2: Find the area of one triangular segment.
Each triangular segment is formed by two consecutive sides of the octagon and the radius of the circle. To find the area of a triangle, we can use the formula:

Area = (1/2) * base * height

In this case, the base of the triangle is the side length of the octagon, and the height can be found using the Pythagorean theorem. The height is the radius of the circle, which is 1 cm, and the hypotenuse is the side length of the octagon.

Using the Pythagorean theorem: (side length)^2 = (height)^2 + (radius)^2

(side length)^2 = (1 cm)^2 + (1 cm)^2
(side length)^2 = 2 cm^2
side length = √2 cm

Therefore, the area of each triangular segment is:
Area = (1/2) * (side length) * (height)
Area = (1/2) * (√2 cm) * (1 cm)
Area = (√2 / 2) cm^2

Step 3: Calculate the total area of the octagon.
Now that we know the area of one triangular segment, we can multiply it by the number of sides (8) to get the total area of the octagon.

Total Area = (Area of one triangular segment) * (Number of sides)
Total Area = (√2 / 2) cm^2 * 8
Total Area = 4√2 cm^2

Therefore, the area of the regular octagon inscribed in a circle with a radius of 1 cm is 4√2 cm^2.