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 #2 #, its base has sides of length #1 #, 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 #2  #, its base has sides of length #1 #, and its base has a corner with an angle of #(3 pi)/8   #.  What is the pyramid's surface area?

Mason Adams · Accepted Answer

T S A = 5.0472

AB = BC = CD = DA = a = 1
Height OE = h = 2
OF = a/2 = 1/2 = 0.5
# EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(2^2+0.5^2) = color(red)(2.0616)#

Area of #DCE = (1/2)*a*EF = (1/2)*1*2.0616 = color(red)(1.0308)#
Lateral surface area #= 4*Delta DCE = 4*1.0308 = color(blue)(4.1232#)#

#/_C = (3pi)/8, /_C/2 = (3pi)/16#
diagonal #AC = d_1# & diagonal #BD = d_2#
#OB = d_2/2 = BCsin (C/2)=1sin((3pi)/16)= 0.5556

#OC = d_1/2 = BC cos (C/2) = 1* cos ((3pi)/16) = 0.8315

Area of base ABCD #= (1/2)*d_1*d_2 = (1/2)(2*0.5556) (2*0.8315) = color (blue)(0.924)#

T S A #= Lateral surface area + Base area#
T S A # =4.1232 + 0.924 = color(purple)(5.0472)#

Carson Arnold · Answer

undefined To find the surface area of the pyramid, we first need to calculate the area of the base and then add the area of the four triangular faces.

1. Calculate the area of the base:
   - The area of a rhombus can be calculated using the formula: $ Area_{rhombus} = \frac{1}{2} \times d_1 \times d_2 $, where $ d_1 $ and $ d_2 $ are the lengths of the diagonals.
   - Since the base has sides of length 1, the diagonals of the rhombus are also 1. Therefore, $ d_1 = d_2 = 1 $.
   - $ Area_{rhombus} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2} $.

2. Calculate the area of one triangular face:
   - The area of a triangle can be calculated using the formula: $ Area_{triangle} = \frac{1}{2} \times base \times height $.
   - The base of each triangular face is one of the sides of the rhombus, which has a length of 1.
   - The height of the triangular face is equal to the height of the pyramid, which is given as 2.
   - $ Area_{triangle} = \frac{1}{2} \times 1 \times 2 = 1 $.

3. Since there are four triangular faces, the total area of all the triangular faces is $ 4 \times Area_{triangle} = 4 $.

4. Finally, calculate the total surface area of the pyramid:
   - $ Surface \: Area = Base \: Area + Total \: Area \: of \: Triangular \: Faces $.
   - $ Surface \: Area = Area_{rhombus} + 4 $.
   - $ Surface \: Area = \frac{1}{2} + 4 = \frac{9}{2} $.

Therefore, the surface area of the pyramid is $ \frac{9}{2} $ square units.