BTGmoderatorDC wrote: ↑

Sun Jan 26, 2020 10:17 pm

A pyramid with four equal-sized flat surfaces and a base of 36 ft2 has a height of 10 feet. What is the total surface area of the pyramid, excluding the base?

A. 12 31^1/2
B. 12 109^1/2
C. 24 31^1/2
D. 24 109^1/2
E. 48 31^1/2

OA B

Source: Veritas Prep

Since the pyramid has four equal-sized flat surfaces, the base must be a square; given that the area of the base is 36 sq feet, the side of the square = √36 = 6 ft.

Note that each of the four equal-sized flat surfaces would be like an isosceles triangle, whose base is 6 ft. We must get the height of the flat surfaces to get their area. To get the height, we must first get the height of the pyramid.

Drop a perpendicular from the tip of the pyramid to its base; from the point where the perpendicular touches the base, drop another perpendicular to one of the sides of the base. It is given that the height of the pyramid = 10 ft; since the side of the base = 6 ft, the length of the perpendicular to the side the base = 6/2 = 3 ft

Thus, this construction forms a rightangled triangle, whose height = 10 ft and base = 3 ft and hypotenuse = height of the flat surface of the pyramid = √(10^2 + 3^2) = √109

Thus, the area of the flat surface = 1/2*height*base = 1/2*√109*6 = 3*√109

Thus, the area of four equal-sized flat surfaces = 4*3*√109 = 12 √109 sq ft

The correct answer: B

Hope this helps!

-Jay
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BTGmoderatorDC wrote: ↑

Sun Jan 26, 2020 10:17 pm

A pyramid with four equal-sized flat surfaces and a base of 36 ft2 has a height of 10 feet. What is the total surface area of the pyramid, excluding the base?

A. 12 31^1/2
B. 12 109^1/2
C. 24 31^1/2
D. 24 109^1/2
E. 48 31^1/2


OA B

Source: Veritas Prep

The total surface area of the pyramid, excluding the base, is also known as the lateral surface area of the pyramid. It is the total area of the 4 triangular faces of the pyramid.

Since the base of the pyramid is 36 ft^2, a side of the base is 6 ft. Recall that the height of the pyramid, half of a side of the base and the slant height of the pyramid form the three sides of a right triangle where the slant height is the hypotenuse. Since the height of the pyramid is 10 ft, half of a side of the base is 3 ft, the slant height, s, can be determined by the equation:

10^2 + 3^2 = s^2

109 = s^2

s = 109^½

Recall that the slant height of the pyramid is also the height of the triangular faces of the pyramid, and a side of the base of the pyramid is the base of the triangular faces. So a triangular face has height = 109^½ and base = 6 and hence an area of 1/2 x 6 x 109^½ = 3 x 109^½. Since there are 4 triangular faces, the total area is 4 x 3 x 109^½ = 12 x 109^½.

Answer: B