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 #4 #, its base's sides have lengths of #3 #, and its base has a corner with an angle of #(5 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 #4  #, its base's sides have lengths of #3 #, and its base has a corner with an angle of #(5 pi)/6 #.  What is the pyramid's surface area?

Sadie Adams · Accepted Answer

T S A = 30.132

AB = BC = CD = DA = a = 3
Height OE = h = 4
OF = a/2 = 3/2 = 1.5
# EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(4^2+1.5^2) = color(red)(4.272)#

Area of #DCE = (1/2)*a*EF = (1/2)*3*4.272 = color(red)(6.408)#
Lateral surface area #= 4*Delta DCE = 4*6.408 = color(blue)(25.632)#

#/_C = pi - (5pi)/6 = (pi)/6#
Area of base ABCD #= a* a * sin /_C = 3^2 sin (pi/6) = 4.5#

T S A #= Lateral surface area + Base area#
T S A # =25.632 + 4.5 = color(purple)(30.132)#

Carson Arnold · Answer

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

First, let's find the area of the rhombus base using the formula for the area of a rhombus:
$$ \text{Area of rhombus} = \frac{d_1 \times d_2}{2} $$
Where $ d_1 $ and $ d_2 $ are the diagonals of the rhombus.

Since the sides of the rhombus have lengths of 3, the diagonals can be calculated as follows:
$$ d_1 = 3 \times \sin\left(\frac{5\pi}{6}\right) $$
$$ d_2 = 3 \times \cos\left(\frac{5\pi}{6}\right) $$

Next, calculate the areas of the triangular faces. Since the height of the pyramid is given as 4, and the base sides are 3, the area of each triangular face can be found using the formula for the area of a triangle:
$$ \text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} $$

Now, find the surface area by summing the area of the base and the areas of the four triangular faces.

After calculating these values, add the area of the base to the total area of the four triangular faces to find the surface area of the pyramid.