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Published byCharleen Rice Modified over 6 years ago

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Example 1 Explain how you could find the area of the regular hexagon shown.

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**Regular Inscribed Polygon**

The diagram shows a regular polygon inscribed in a circle. Center of circle = center of the polygon Radius of circle = radius of the polygon

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**Regular Inscribed Polygon**

The apothem of the polygon is the distance from the center to any side of the polygon. Apothem = height of isosceles triangle with 2 radii as legs

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**Regular Inscribed Polygon**

A central angle of a polygon is an angle formed by two consecutive radii. Measure of central angle =

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**Areas of Regular Polygons Perimeter and Area of Similar Figures**

Objective: To find the area of a regular n-gon To describe the effects on perimeter and area when dimensions are changed proportionally

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Example 2 Identify the center, a radius, an apothem, and a central angle of the polygon. Find m<XPY, m<XPQ, m<PXQ.

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Example 3 Assume a regular n-gon has a side length of s and an apothem of a. Find a formula for the area of the regular n-gon.

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**Area of a Regular Polygon**

The area of a regular n-gon with side length s is half the product of the apothem a and the perimeter P.

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Regular 3-gon What is the measure of each central angle in an equilateral triangle? What is the measure of the angle formed by the apothem and the radius of the triangle?

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Regular 4-gon What is the measure of each central angle in a square? What is the measure of the angle formed by the apothem and the radius of a square?

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Regular 5-gon What is the measure of each central angle in a regular pentagon? What is the measure of the angle formed by the apothem and the radius of the pentagon?

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Regular 6-gon What is the measure of each central angle in a regular hexagon? What is the measure of the angle formed by the apothem and the radius of the hexagon?

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Example 4 Find the area of each regular polygon.

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Summary

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Example 5 Find the area of each regular polygon. A =

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Example 6 Find the area of each regular polygon. A =

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Example 7 Find a formula for the area of a regular hexagon in terms of s, the side length.

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Example 8 The perimeter of a regular hexagon is 48 cm. What is the area of the hexagon?

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Example 9 Find the area of the shaded region.

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Example 10 Rectangle ABCD ~ PQRS with a scale factor of 3:4. Find the perimeter and area of rectangle PQRS.

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**Perimeter of Similar Polygons**

If two polygons are similar with the lengths of corresponding sides in the ratio of a:b, then the ratio of their perimeters is a:b.

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**Area of Similar Polygons**

If two polygons are similar with the lengths of corresponding sides in the ratio of a:b, then the ratio of their areas is a2:b2.

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**Example 11 In the diagram ΔABC ~ ΔDEF. Find the indicated ratio.**

Ratio (red to blue) of the perimeters Ratio (red to blue) of the areas

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Example 12 Stuart is installing the same carpet in a bedroom and den. The floors of the rooms are similar. The carpet for the bedroom costs $117. Carpet is sold by the square foot. How much does it cost to carpet the den?

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Example 13 The polygons below are similar. Find the values of x and y.

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