Regular Polygons with Rational Area or Perimeter.
Write an expression that finds the area of one triangle in an n-gon, using a to represent the apothem and s to represent a side of the n-gon. Then use the expression to write a formula that finds the area of the entire n-gon, where n represents the number of sides of the n-gon. Explain how to determine the perimeter of a polygon, P.
Question 194497This question is from textbook: Write the perimeter formula of a rectangle, and then substitute the following conditions into the formula: The perimeter of a rectangular garden is 100 feet, and the length of the garden in 12 feet more than the width. Use x to represent the width. This question is from textbook Answer by doukungfoo(195) (Show Source).
Formula for the area of a regular polygon. 2. Given the radius (circumradius) If you know the radius (distance from the center to a vertex, see figure above): where r is the radius (circumradius) n is the number of sides sin is the sine function calculated in degrees (see Trigonometry Overview). To see how this equation is derived, see Derivation of regular polygon area formula.
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Given the radius r. The radius of a regular polygon is the distance from the center to any vertex. In the figure below the leg of the isosceles triangle is a radius r of the polygon. We add a perpendicular h from the apex to the base. In this case, let t be the whole angle at the apex. From the figure we see that The area of any triangle is half the base times height, and since x is already.
Another interesting thought suggested by Ferry’s puzzle is to give the exact formula for the expected area of an n-sided convex polygon whose vertices are randomly selected points on the unit circle (assuming a uniform distribution on the perimeter). To answer this question we obviously can't make use of the small-angle assumption, so we must use the full sine formulas.
Students will deduce the general expressions for perimeter and area of an n-sided polygon based on the previous lessons. Students will understand the concept of representing the number of sides of a regular polygon with the variable n. Procedure: Perimeter. 1. Collectively recall the various expressions discovered from the previous lessons.