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The base of a regular pentagonal pyramid has a perimeter of 60 feet and an area of 248 square feet. The slant height of the pyramid is 9 feet. Find the surface area of the pyramid.

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Answer:

425.45 square feet

Explanation:

Surface Area = Base Area + Lateral Area

Base Area = (5/4) * s^2 * cot(π/5)

Lateral Area = (1/2) * P * h

Perimeter of a regular pentagon:

P = 5s

Lateral Area of a pyramid:

Lateral Area = (1/2) * P * h

Area of a regular pentagon:

Base Area = (5/4) * s^2 * cot(π/5)

where:

- P represents the perimeter of the base

- s represents the length of each side of the base

- h represents the slant height of the pyramid

Sure! Here's a breakdown of the calculations:

Perimeter of the base:

P = 60 feet

Length of each side of the base:

s = P / 5

s = 60 / 5

s = 12 feet

Slant height:

h = 9 feet

Lateral Area:

Lateral Area = (1/2) * P * h

Lateral Area = (1/2) * 60 * 9

Lateral Area = 270 square feet

Base Area:

Base Area = (5/4) * s^2 * cot(π/5)

Base Area = (5/4) * 12^2 * cot(π/5)

Base Area ≈ 155.45 square feet

Surface Area:

Surface Area = Lateral Area + Base Area

Surface Area ≈ 270 + 155.45

Surface Area ≈ 425.45 square feet

Therefore, the surface area of the regular pentagonal pyramid is approximately 425.45 square feet.

To find the surface area of the regular pentagonal pyramid, we need to calculate the lateral area (excluding the base) and then add it to the base area.

The perimeter of the base is given as 60 feet, and since the base is a regular pentagon, each side has the same length. Let's denote the length of each side of the pentagon as "s."

The formula to calculate the perimeter of a regular pentagon is P = 5s, where P represents the perimeter. In this case, we have P = 60 feet. Therefore, we can find the length of each side:

5s = 60

s = 60 / 5

s = 12 feet

Now, we know that the slant height of the pyramid is 9 feet. The slant height represents the height of each triangular face on the lateral surface of the pyramid.

The lateral area of a pyramid can be calculated using the formula: Lateral Area = (1/2) * Perimeter * Slant Height.

Plugging in the given values, we can calculate the lateral area:

Lateral Area = (1/2) * 60 * 9

Lateral Area = 270 square feet

Now, let's find the base area of the pentagon. The area of a regular pentagon can be calculated using the formula: Base Area = (5/4) * s^2 * cot(π/5), where s represents the length of each side.

Plugging in the given values, we can calculate the base area:

Base Area = (5/4) * 12^2 * cot(π/5)

Base Area ≈ 155.45 square feet

Finally, we can find the surface area by adding the lateral area to the base area:

Surface Area = Lateral Area + Base Area

Surface Area ≈ 270 + 155.45

Surface Area ≈ 425.45 square feet

Therefore, the surface area of the regular pentagonal pyramid is approximately 425.45 square feet

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User Janne Karila
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