AboutPaul Klarreich Expertise I can answer questions in basic to advanced algebra (theory of equations, complex numbers), precalculus (functions, graphs, exponential, logarithmic, and trigonometric functions and identities), basic probability, and finite mathematics, including mathematical induction.
I can also try (but not guarantee) to answer questions on Abstract Algebra
-- groups, rings, etc. and Analysis -- sequences, limits, continuity.
I won't understand specialized engineering or business jargon.
Experience I taught at a two-year college for 25 years, including all subjects from algebra to third-semester calculus.
Expert: Paul Klarreich Date: 5/11/2008 Subject: Trigonometry
Question A regular polygon of n sides is inscribed in a circle of radius r. If the area of the polygon is 2r^2sqrt2, how many sides does it have. I know the answer is 8, but i don;t know how to get there.
Answer Questioner: Patrick
Category: Advanced Math
Private: No
Subject: Trigonometry
Question: A regular polygon of n sides is inscribed in a circle of radius r. If the area of the polygon is 2r^2sqrt2, how many sides does it have. I know the answer is 8, but i don;t know how to get there.
..............................................
Hi, Patrick,
Take your polygon and chop it up into n triangles. Each one looks like this:
/\
/ \
r/ \r
/ \
/ \
-----------
There is this formula for the area of a triangle:
Area = 1/2 ab sin C, where a and b are two sides and C is the angle between them.
In this case, a = b = r, and C = 360/n, the central angle.
Area of one triangle = 1/2 r^2 sin 360/n
Area of n triangles = n/2 r^2 sin 360/n
Now sin 45 = sqrt(2)/2, so it appears 360/n = 45, so n = 360/5 = 8.
Putting that in, we have:
Area = 8/2 r^2 sin 360/8 = 4 r^2 sin(45) = 4 r^2 sqrt(2)/2 =
2 r^2 sqrt(2), which is what you have.
