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 #8 #, its base has sides of length #4 #, and its base has a corner with an angle of #(3 pi)/8 #. 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 #8  #, its base has sides of length #4 #, and its base has a corner with an angle of #(3 pi)/8   #.  What is the pyramid's surface area?

Claire Akin · Accepted Answer

T S A = 80.7519

AB = BC = CD = DA = a = 4
Height OE = h = 8
OF = a/2 = 1/2 = 2
# EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(8^2+2^2) = color(red)(8.2462)#

Area of #DCE = (1/2)*a*EF = (1/2)*4*8.2462 = color(red)(16.4924)#
Lateral surface area #= 4*Delta DCE = 4*16.4924 = color(blue)(65.9696)#

#/_C = (3pi)/8, /_C/2 = (3pi)/16#
diagonal #AC = d_1# & diagonal #BD = d_2#
#OB = d_2/2 = BCsin (C/2)=4sin((3pi)/16)= 2.2223

#OC = d_1/2 = BC cos (C/2) = 4* cos ((3pi)/16) = 3.3259

Area of base ABCD #= (1/2)*d_1*d_2 = (1/2)(2*2.2223) (2*3.3259) = color (blue)(14.782)#

T S A #= Lateral surface area + Base area#
T S A # =65.9696 + 14.7823 = color(purple)(80.7519)#

Parker Allen · Answer

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

First, let's find the area of one of the triangular faces:
Area of a triangle = 1/2 * base * height

In this case, the base of the triangle is one of the sides of the rhombus base, which is 4 units, and the height of the triangle is the height of the pyramid, which is 8 units. Therefore:
Area of one triangular face = 1/2 * 4 * 8 = 16 square units

Since the pyramid has four identical triangular faces, the total area of the triangular faces is:
Total area of triangular faces = 4 * 16 = 64 square units

Next, let's find the area of the rhombus base. The area of a rhombus can be calculated using the formula:
Area of a rhombus = (diagonal1 * diagonal2) / 2

The diagonals of the rhombus can be calculated using trigonometry. Given that one corner angle of the rhombus is (3π)/8, we can use the law of sines to find the lengths of the diagonals. Let d1 and d2 be the lengths of the diagonals:
d1 = 2 * (4/2) * sin(π/8)
d2 = 2 * (4/2) * sin(π/8)

Now, we can calculate the area of the rhombus using the formula:
Area of rhombus = (d1 * d2) / 2

After finding the area of the rhombus and the total area of the triangular faces, we can sum them up to get the total surface area of the pyramid:
Total surface area = Total area of triangular faces + Area of rhombus