Question

What is the area of a circumscribed polygon with 3 sides? The circle has a radius of 1 by the way.

Answer 1

also what is the are of a circumscribed polygon with 4 sides? and 6 sides?
Hint: use formula A=1/2ap where a is apothem and p is perimeter.

Answer 2

Join the center of the circle to the vertices of the polygon.
This divides the polygon into n isosceles traingles having equal sides of length 1 with the included angle = 2π/n
So area of the polygon
= n * area of each triangle
= n * (1/2) * (1)^2 * sin(2π/n)
= (n/2) sin(2π/n)
Area for polygon of sides
3 = (3/2)sin(2π/3) = 3√3/4 sq. units
so for 4 and 6 just substitute where n is in the final formula.

Answer 3

like this right? how do we know the sides are length 1? The only number given in the problem is that the circle has a radius of 1.

Answer 4

Not like that circumscribed means polygon is inside circle, look at :
http://en.wikipedia.org/wiki/Circumscribed_circle

Answer 5

if done this way the smaller triangles are 30-60-90 degree triangles right?
Using the 30-60-90 theorem it states that the length of the hypotenuse is twice the length of the short leg, and the length of the long leg is sqrt3 times as long as the length of the short leg