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 #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 #8  #, 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?

Sadie Adams · Accepted Answer

T S A = 198.5344

AB = BC = CD = DA = a = 8
Height OE = h = 8
OF = a/2 = 8/2 = 4
# EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(8^2+4^2) = color(red)(8.9443)#

Area of #DCE = (1/2)*a*EF = (1/2)*8*8.9443 = color(red)(35.7772)#
Lateral surface area #= 4*Delta DCE = 4*35.7772 = color(blue)(143.1088)#

#/_C = pi/3, /_C/2 = pi/6#
diagonal #AC = d_1# & diagonal #BD = d_2#
#OB = d_2/2 = BCsin (C/2)=8sin(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*4) (2*6.9282) = color (blue)(55.4256)#

T S A #= Lateral surface area + Base area#
T S A # =143.1088 + 55.4256 = color(purple)(198.5344)#

Caroline Aucoin · Answer

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

First, let's calculate the area of one triangular face:

1. The base of each triangle is one side of the rhombus, which has a length of 8.
2. The height of each triangle can be found using trigonometry. Since the angle at one corner of the rhombus is $ \frac{\pi}{3} $, the height of the triangle is $ 8 \times \sin\left(\frac{\pi}{3}\right) = 8 \times \frac{\sqrt{3}}{2} = 4\sqrt{3} $.
3. The area of each triangular face is $ \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 4\sqrt{3} = 16\sqrt{3} $.

Since there are four triangular faces, the total area contributed by the triangular faces is $ 4 \times 16\sqrt{3} = 64\sqrt{3} $.

Now, let's calculate the area of the rhombus base:

1. The area of a rhombus can be calculated using the formula: $ \text{Area} = \frac{1}{2} \times \text{diagonal}_1 \times \text{diagonal}_2 $.
2. The diagonals of the rhombus can be found using trigonometry. The shorter diagonal (across the corners with angle $ \frac{\pi}{3} $) is $ 8 \times \cos\left(\frac{\pi}{3}\right) = 8 \times \frac{1}{2} = 4 $, and the longer diagonal (across the opposite corners) is twice the length of the shorter diagonal, so it is $ 2 \times 4 = 8 $.
3. The area of the rhombus base is $ \frac{1}{2} \times 4 \times 8 = 16 $.

The total surface area of the pyramid is the sum of the areas of the triangular faces and the rhombus base:

$ \text{Surface Area} = 64\sqrt{3} + 16 $.

Calculating the numerical value, we get:

$ \text{Surface Area} \approx 110.85 $.