A pyramid has a base in the shape of a rhombus and a peak directly above the base's center. The pyramid's height is #3 #, its base has sides of length #8 #, and its base has a corner with an angle of # pi/3 #. What is the pyramid's surface area?

Question

A pyramid has a base in the shape of a rhombus and a peak directly above the base's center.  The pyramid's height is #3  #, its base has sides of length #8 #, and its base has a corner with an angle of # pi/3   #.  What is the pyramid's surface area?

Caroline Aucoin · Accepted Answer

T S A #= color(purple)(190.8252)#

AB = BC = CD = DA = a = 8
Height OE = h = 3
OF = a/2 = 8/2 = 4
# EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(3^2+4^2) = color(red)5#

Area of #DCE = (1/2)*a*EF = (1/2)*8*5 = color(red)(20)#
Lateral surface area #= 4*Delta DCE = 4*20 = color(blue)(80)#

#/_C = pi/3, /_C/2 = pi/6#
diagonal #AC = d_1# & diagonal #BD = d_2#
#OB = d_2/2 = BC*sin (C/2)=8*sin(pi/6)= **4**#

#OC = d_1/2 = BC cos (C/2) = 8* cos (pi/6) = 6.9282

Area of base ABCD #= (1/2)*d_1*d_2 = (1/2)(2*8) (2*6.9282) = color (blue)(110.8512)#

Total Surface Area #= Lateral surface area + Base area#
T S A # =80 + 110.8252 = color(purple)(190.8252)#

Aurora Ayala · Answer

undefined To find the surface area of the pyramid, we need to calculate the areas of the four triangular faces and the rhombus base.

1. The area of each triangular face can be calculated using the formula: Area = (1/2) * base * height. Since the base of each triangular face is a side of the rhombus base and the height is the height of the pyramid, we have:
   Area of each triangular face = (1/2) * 8 * 3 = 12 square units.

2. The area of the rhombus base can be calculated using the formula: Area = base * height. Since the rhombus has diagonals of equal length, and the angle between adjacent sides is π/3, each diagonal can be calculated using the formula: diagonal = side * √(2 + 2 * cos(angle)).
   The side length of the rhombus is 8, and the angle is π/3, so:
   Diagonal = 8 * √(2 + 2 * cos(π/3)) = 8 * √(2 + 2 * (1/2)) = 8 * √3.
   The area of the rhombus base is then:
   Area = diagonal1 * diagonal2 / 2 = (8 * √3) * (8 * √3) / 2 = 96 square units.

3. The total surface area of the pyramid is the sum of the areas of the triangular faces and the base:
   Surface area = 4 * Area of each triangular face + Area of the base = 4 * 12 + 96 = 48 + 96 = 144 square units.

Therefore, the surface area of the pyramid is 144 square units.