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's sides have lengths of #5 #, and its base has a corner with an angle of #( pi)/6 #. 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's sides have lengths of #5 #, and its base has a corner with an angle of #( pi)/6 #.  What is the pyramid's surface area?

Avery Allen · Accepted Answer

T S A = 20.5387

AB = BC = CD = DA = a = 5
Height OE = h = 2
OF = a/2 = 1/2 = 2.5
# EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(2^2+2.5^2) = color(red)(3.2016)#

Area of #DCE = (1/2)*a*EF = (1/2)*5*3.2016 = color(red)(2.5125)#
Lateral surface area #= 4*Delta DCE = 4*2.5125 = color(blue)(8.04)#

#/_C = pi/6, /_C/2 = pi/12#
diagonal #AC = d_1# & diagonal #BD = d_2#
#OB = d_2/2 = BCsin (C/2)=5sin(pi/12)= 1.294

#OC = d_1/2 = BC cos (C/2) = 5* cos (pi/12) = 4.8295

Area of base ABCD #= (1/2)*d_1*d_2 = (1/2)(2*1.294) (2*4.8295) = color (blue)(12.4987)#

T S A #= Lateral surface area + Base area#
T S A # =8.04 + 12.4987 = color(purple)(20.5387)#

Theodore Abad · Answer

undefined To find the surface area of the pyramid, we first need to calculate the area of each face and then sum them up.

1. Calculate the area of the rhombus base:
   The area of a rhombus can be found using the formula: Area = base * height.
   Since the base's sides have lengths of 5, and one angle of the rhombus is π/6, we can use trigonometry to find the height of the rhombus.
   The height (h) of the rhombus can be calculated as: h = side * sin(angle).
   Substituting the given values, we get: h = 5 * sin(π/6) ≈ 5 * 0.5 = 2.5.
   So, the area of the rhombus base is: Area_base = 5 * 2.5 = 12.5 square units.

2. Calculate the area of each triangular face:
   Each triangular face has a base length equal to the side length of the rhombus (5 units) and a height equal to the height of the pyramid (2 units).
   The area of each triangular face is: Area_triangle = (1/2) * base * height.
   Substituting the given values, we get: Area_triangle = (1/2) * 5 * 2 = 5 square units for each face.

3. Since there are four triangular faces, the total area of all triangular faces is: Total_area_triangles = 4 * Area_triangle = 4 * 5 = 20 square units.

4. The total surface area of the pyramid is the sum of the area of the base and the total area of the triangular faces:
   Total_surface_area = Area_base + Total_area_triangles = 12.5 + 20 = 32.5 square units.

Therefore, the surface area of the pyramid is 32.5 square units.