10-3 Areas of Regular Polygons

Quick Review

The center of a regular polygon C is the center of its circumscribed circle. The radius r is the distance from the center to a vertex. The apothem a is the perpendicular distance from the center to a side. The area of a regular polygon with apothem a and perimeter p is A = 1 2 a p . eh equals . 1 half , eh p .

Example

What is the area of a hexagon with apothem 17.3 mm and perimeter 120 mm?

A = 1 2 a p Use the area formula .   = 1 2 ( 17.3 ) ( 120 ) = 1038 Substitute and simplify . table with 2 rows and 3 columns , row1 column 1 , eh , column 2 equals , 1 half , eh p , column 3 cap usetheareaformula . . , row2 column 1 , , column 2 equals , 1 half . open , 17.3 , close . open 120 close , equals , 1038 , column 3 cap substituteandsimplify . . , end table

The area of the hexagon is 1038   m   m 2 . 1038 , m , m squared , .

Exercises

Find the area of each regular polygon. If your answer is not an integer, leave it in simplest radical form.

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17. What is the area of a regular hexagon with a perimeter of 240 cm?
18. What is the area of a square with radius 7.5 m?

Sketch each regular polygon with the given radius. Then find its area to the nearest tenth.

19. triangle; radius 4 in.
20. square; radius 8 mm
21. hexagon; radius 7 cm

10-4 Perimeters and Areas of Similar Figures

Quick Review

If the scale factor of two similar figures is a b , eh over b , comma  then the ratio of their perimeters is a b , eh over b , comma  and the ratio of their areas is a 2 b 2 fraction eh squared , over b squared end fraction

Example

If the ratio of the areas of two similar figures is 4 9 , 4 ninths , comma  what is the ratio of their perimeters?

Find the scale factor.

4 9 = 2 3 fraction square root of 4 , over square root of 9 end fraction , equals , 2 thirds  Take the square root of the ratio of areas.

The ratio of the perimeters is the same as the ratio of corresponding sides, 2 3 . 2 thirds , .

Exercises

For each pair of similar figures, find the ratio of the area of the first figure to the area of the second.

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26. If the ratio of the areas of two similar hexagons is 8 : 25, what is the ratio of their perimeters?

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