consider a circle of radius 1, and corresponding inscribed and circumscribed polygons with the number of sides n = 3, 4, 5, 6, and 8.
A: For each n = 3, 4, 5, 6 & 8, what are the areas of the inscribed and circumscribed polygons with n sides?
divide each polygon into identical isosceles triangles. Find the area of one triangle and multiply by the number of sides.
To find the areas of the inscribed and circumscribed polygons with n sides, we need to use some basic geometric formulas.
1. Inscribed Polygon:
An inscribed polygon is a polygon that has its vertices lying on the circumference of a circle. In this case, the circle has a radius of 1.
To find the area of an inscribed polygon, we can use the formula:
Area of inscribed polygon = (n * s^2) / (4 * tan(π/n))
where n is the number of sides and s is the length of each side.
2. Circumscribed Polygon:
A circumscribed polygon is a polygon that has its sides tangential to the circumference of a circle. In this case, the circle has a radius of 1.
To find the area of a circumscribed polygon, we can use the formula:
Area of circumscribed polygon = (n * s^2) / (4 * tan(π/n))
Again, n is the number of sides and s is the length of each side.
Now, we can calculate the areas of the inscribed and circumscribed polygons for each given value of n.
For n = 3 (triangle):
Area of inscribed triangle = (3 * s^2) / (4 * tan(π/3))
Area of circumscribed triangle = (3 * s^2) / (4 * tan(π/3))
For n = 4 (square):
Area of inscribed square = (4 * s^2) / (4 * tan(π/4))
Area of circumscribed square = (4 * s^2) / (4 * tan(π/4))
For n = 5 (pentagon):
Area of inscribed pentagon = (5 * s^2) / (4 * tan(π/5))
Area of circumscribed pentagon = (5 * s^2) / (4 * tan(π/5))
For n = 6 (hexagon):
Area of inscribed hexagon = (6 * s^2) / (4 * tan(π/6))
Area of circumscribed hexagon = (6 * s^2) / (4 * tan(π/6))
For n = 8 (octagon):
Area of inscribed octagon = (8 * s^2) / (4 * tan(π/8))
Area of circumscribed octagon = (8 * s^2) / (4 * tan(π/8))
To find the values of s for each polygon, we can consider the fact that the radius of the circle is 1. For example, in an inscribed equilateral triangle, each side (s) would be the same as the radius, which is 1.
By substituting the values of n and s into the formulas, we can calculate the areas of the inscribed and circumscribed polygons for each given value of n.