What is the formula for finding the area of a regular polygon with perimeter [tex]$P$[/tex] and apothem length [tex]$a$[/tex]?
A. [tex]$A=P a$[/tex]
B. [tex]$a=1 / 2(P A)$[/tex]
C. [tex]$a=P A$[/tex]
D. [tex]$A=1 / 2(P a)$[/tex]
Answer (2)
The problem requires identifying the correct formula for the area of a regular polygon.
The area A of a regular polygon is given by A = 2 1 a P , where a is the apothem and P is the perimeter.
Comparing this with the given options, the correct formula is A = 2 1 ( P a ) .
Therefore, the answer is: A = 2 1 ( P a ) .
Explanation
Problem Analysis The question asks for the formula for the area A of a regular polygon with perimeter P and apothem length a . We need to identify the correct formula from the given options.
Recall the Area Formula The area of a regular polygon can be found using the formula:
A = 2 1 a P
where: A is the area of the regular polygon, a is the length of the apothem (the perpendicular distance from the center of the polygon to the midpoint of a side), P is the perimeter of the polygon.
Compare with Given Options Now, let's compare this formula with the given options:
A. A = P a (Incorrect, missing the 2 1 factor) B. a = 2 1 ( P A ) (Incorrect, this is not the standard area formula) C. a = P A (Incorrect, this is not the standard area formula) D. A = 2 1 ( P a ) (Correct, matches the standard area formula)
Final Answer The correct formula for the area of a regular polygon with perimeter P and apothem length a is:
A = 2 1 P a
Examples
Imagine you're designing a mosaic pattern using regular hexagonal tiles. Each tile has a perimeter of 30 cm and an apothem of approximately 4.33 cm. To find the area of each tile, you can use the formula A = 2 1 P a . Plugging in the values, you get A = 2 1 × 30 × 4.33 = 64.95 cm 2 . This helps you calculate the amount of material needed for each tile, ensuring you have enough to complete your mosaic project.
The formula for the area of a regular polygon with perimeter P and apothem a is A = 2 1 P a . Therefore, option D is the correct answer. This formula helps in calculating the area by relating the apothem and perimeter of the polygon.
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