What is the lateral area of a regular pentagonal pyramid that has a slant height of 18 in. and a base side length of 9 in.?
405 in.^2
810 in.^2
81 in.^2
162 in.^2
I don't know how to do this:/ Please help?
Thanks, I know its 405 now:)
the slant height is the height of the triangles that make up the sides of the pyramid
the base of the triangles is the base side length
pentagonal means five triangular sides with a height of 18 and a base of 9
find the areas and add them up (or find one and multiply by 5)
Well, don't worry! I'm here to assist you with a little bit of humor!
So, we have a regular pentagonal pyramid with a slant height of 18 inches and a base side length of 9 inches. To find the lateral area, we need to calculate the total surface area of all the triangular faces of the pyramid.
First, let's calculate the area of one triangular face. We can divide the pentagon into five congruent triangles by drawing line segments from each of the vertices to the center of the base. This creates five isosceles triangles.
The slant height of each triangle is 18 inches, which forms the height of the triangle, and the base of each triangle is 9 inches. We can find the area of one triangle using the formula:
A = (1/2) * base * height
A = (1/2) * 9 in * 18 in
Now, let's do the math:
A = 81 in^2
Since there are five triangles in total, we multiply the area of one triangle by 5 to find the lateral area of the pyramid.
Lateral Area = 81 in^2 * 5
Lateral Area = 405 in^2
So, the correct answer is 405 in^2. Ta-da!
To find the lateral area of a regular pentagonal pyramid, you need to know the slant height and the base side length. The lateral area is the sum of the areas of the triangular faces that make up the sides of the pyramid.
First, calculate the apothem of the base, which is the distance from the center of the base to the midpoint of any side. In a regular pentagon, the apothem can be found using the formula:
apothem = (side length) / (2 * tan(180° / number of sides))
In this case, the side length is 9 inches, and the number of sides is 5 (since it is a pentagon). Plugging in the values:
apothem = (9 in) / (2 * tan(180° / 5))
= (9 in) / (2 * tan(36°))
≈ 5.89 in
Next, calculate the slant height of the triangular faces. This can be done using Pythagoras' theorem, where the height is the apothem and the base is half the base side length of the pentagon:
slant height = sqrt((apothem^2) + ((base side length / 2)^2))
= sqrt((5.89 in)^2 + ((9 in / 2)^2))
≈ 9.96 in
Now, calculate the area of each triangular face using the formula for the area of a triangle:
area of triangle = (base * height) / 2
= (9 in * 9.96 in) / 2
≈ 44.82 in^2
Since there are 5 triangular faces, multiply the area of one face by 5 to get the total lateral area:
lateral area = 5 * (44.82 in^2)
≈ 224.10 in^2
Rounded to the nearest whole number, the lateral area of the regular pentagonal pyramid is 224 in^2.
To find the lateral area of a regular pentagonal pyramid, you need to find the area of each triangular face and then sum them up.
First, let's find the area of one of the triangular faces. Since the base of the pyramid is a regular pentagon, each face is an isosceles triangle. To calculate the area of an isosceles triangle, you need to know the length of the base and the height.
In this case, the side length of the base is given as 9 inches. To find the height, we can divide the isosceles triangle into two right-angled triangles by drawing a perpendicular line from the apex (top vertex) to the base. This perpendicular line is also the height of the triangle.
Since the triangle is isosceles, the perpendicular line will bisect the base, creating two congruent right-angled triangles. We can use Pythagoras' theorem to find the height.
Using Pythagoras' theorem:
hypotenuse^2 = base^2 + height^2
Given hypotenuse (slant height) = 18 inches and base = 9 inches
18^2 = 9^2 + height^2
324 = 81 + height^2
height^2 = 243
height = √243 ≈ 15.59 inches
Now that we have the base and height, we can find the area of one triangular face using the formula: Area = (base * height) / 2
Area = (9 * 15.59) / 2 ≈ 70.15 square inches
Since the regular pentagonal pyramid has five identical triangular faces, you need to multiply the area of one triangular face by 5 to get the lateral area.
Lateral Area = 70.15 * 5 = 350.75 square inches.
Therefore, the lateral area of the given regular pentagonal pyramid is approximately 350.75 square inches.
However, none of the answer choices match the calculated result. Please double-check the given answer options or confirm the dimensions provided.