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 #5 #, its base's sides have lengths of #7 #, 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 #5  #, its base's sides have lengths of #7 #, and its base has a corner with an angle of #(5 pi)/6 #.  What is the pyramid's surface area?

Caroline Aucoin · Accepted Answer

T S A = 108.9464

AB = BC = CD = DA = a = 7
Height OE = h = 5
OF = a/2 = 7/2 = 3.5
# EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(5^2+(7/2)^2) = color(red)(6.1033)#

Area of #DCE = (1/2)*a*EF = (1/2)*7*6.1033 = color(red)(21.3616)#
Lateral surface area #= 4*Delta DCE = 4*21.3616 = color(blue)(84.4464)#

#/_C = (pi) - ((5pi)/6) = (pi)/6#
Area of base ABCD #= a* a * sin /_C = 7^2 sin (pi/6) = 24.5#

T S A #= Lateral surface area + Base area#
T S A # =84.4464 + 24.5 = color(purple)(108.9464)#

Autumn Armstrong · Answer

undefined To find the surface area of the pyramid, first, calculate the area of the rhombus base using the formula: $ A_{base} = \frac{d_1 \times d_2}{2} $, where $ d_1 $ and $ d_2 $ are the diagonals of the rhombus.

Given that the side length of the rhombus is $ 7 $, and one angle of the rhombus is $ \frac{5\pi}{6} $, we can find the diagonals using trigonometric relationships in a rhombus.

The diagonals of a rhombus can be calculated as follows:
$$ d_1 = 2 \times 7 \times \sin(\frac{5\pi}{12}) $$
$$ d_2 = 2 \times 7 \times \sin(\frac{\pi}{3}) $$

Then, calculate the height of each triangular face using the Pythagorean theorem:
$$ h = \sqrt{5^2 - \left(\frac{7}{2}\right)^2} $$

The surface area of the pyramid can be found by adding the areas of the four triangular faces (each having a base equal to the side length of the rhombus and height calculated above) and the area of the base rhombus.

Finally, add up the areas to find the total surface area of the pyramid.