Final answer:
Radius = 7.43 cm / cos(180/5) ≈ 9.09 cm
Step-by-step explanation:
To find the length of a side of a regular pentagon with an apothem of 7.43 cm, we need to understand the relationship between a regular pentagon's apothem, side length, and radius of the circumscribed circle. The apothem of a regular polygon is a line from the center to the middle of one of its sides. This line is perpendicular to that side and forms two congruent right triangles within the pentagon. We can use trigonometry to solve for the side length, using the fact that the central angle of each triangle is 360°/5 (since there are five segments) or 72°. Since the apothem forms the right triangle's height, we can look at half the central angle which is 36° to find the side length.
The length of the side of the pentagon (s) can be found using the formula s = 2 • apothem • tan(36°). Plugging in the given apothem of 7.43 cm, s = 2 • 7.43 cm • tan(36°). To find the radius (R) of the circumscribed circle, we can use the formula R = apothem • (1/cos(36°)). Both results should be rounded to the nearest hundredth as the question asks.
Finally, the perimeter and area of this pentagon can be found accordingly. The perimeter is 5 times the side length, and the area can be calculated with the formula A = 1/2 • perimeter • apothem, using the found side length and the given apothem.
To find the length of a side of the pentagon, we can use the formula:
Side length = 2 x apothem x tan(180/n)
where n is the number of sides of the polygon. In this case, n = 5. Plugging in the values, we get:
Side length = 2 x 7.43 cm x tan(180/5) ≈ 14.40 cm
To find the length of the radius of the circumscribed circle, we can use the formula:
Radius = apothem / cos(180/n)
Plugging in the values, we get:
Radius = 7.43 cm / cos(180/5) ≈ 9.09 cm