Find the surface area of the regular pyramid shown to the nearest whole number the figure is not drawn to scale
17m
8.5 √3m ---
9m
To find the surface area of a regular pyramid, we need to calculate the sum of the areas of all its faces.
1. The base of the pyramid is a regular triangle, so let's calculate its area. The formula for the area of a triangle is (1/2) * base * height. Since the base of the triangle is 17m and the height can be obtained from the Pythagorean theorem, we have:
height^2 = (8.5√3/2)^2 + 9^2
height^2 = (72.25/4) + 81
height^2 = 18.0625 + 81
height^2 = 99.0625
height ≈ √99.0625
height ≈ 9.95m
Therefore, the area of the base triangle is (1/2) * 17m * 9.95m = 76.775m².
2. The lateral faces of the pyramid are congruent isosceles triangles. To find their area, we need to calculate the slant height (l) of the pyramid first. Using the Pythagorean theorem again, we have:
l^2 = 9.95^2 + (8.5√3/2)^2
l^2 = 99.025 + 72.25
l^2 ≈ 171.275
l ≈ √171.275
l ≈ 13.1m
Now, we can calculate the area of each lateral face using the formula (1/2) * base * height, where the base is 17m and the height is 13.1m. The area of each lateral face is (1/2) * 17m * 13.1m = 111.35m².
3. The total surface area of the pyramid is the sum of these areas: base area + 4 * lateral face area, since there are 4 lateral faces.
total surface area = 76.775m² + (4 * 111.35m²)
total surface area = 76.775m² + 445.4m²
total surface area ≈ 522.18m²
Therefore, the surface area of the regular pyramid is approximately 522m² to the nearest whole number.
To find the surface area of a pyramid, we need to find the area of each face and then add them together. The base of the pyramid is a regular triangle with side length 17m and height 8.5√3m. We can use the formula for the area of an equilateral triangle:
A = (sqrt(3)/4) * s^2
where s is the length of a side. Plugging in s = 17m, we get:
A = (sqrt(3)/4) * 17^2
A = 165.25m^2
This is the area of one face of the pyramid. There are four other faces, which are all triangles with base 17m and slant height 9m (the height of the pyramid). To find the height of these triangles, we can use the Pythagorean theorem:
a^2 + b^2 = c^2
where a and b are the legs and c is the hypotenuse (slant height). Plugging in a = 8.5√3m and c = 9m, we get:
(8.5√3)^2 + b^2 = 9^2
72.25 + b^2 = 81
b^2 = 8.75
b ≈ 2.96m
So the height of each triangular face is approximately 2.96m. The area of each triangular face is:
A = (1/2) * base * height
A = (1/2) * 17m * 2.96m
A ≈ 25.16m^2
Adding up the areas of all five faces, we get:
165.25m^2 + 4(25.16m^2)
= 273.97m^2
To the nearest whole number, the surface area of the pyramid is 274m^2.