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 #7 #, its base's sides have lengths of #5 #, and its base has a corner with an angle of #(5 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 #7  #, its base's sides have lengths of #5 #, and its base has a corner with an angle of #(5 pi)/8 #.  What is the pyramid's surface area?

Allison Allen · Accepted Answer

Total Surface Area

#color(green)( T S A = A_T = A_L + A_R = 23.097 + 74.3303 = 97.4273)#

Area of Rhombus base #A_R = a * a sin theta#

#A_R = 5 * 5 * sin ((5pi)/8) ~~ 23.097#

Area of slant triangle #A_t = (1/2) a * h_l# where #color(red)(a)# is the rhombus base length and #color(red)(h_l)#is the slant height of the triangle.

#h_l = sqrt(h^2 + (a/2)^2)# where #color(red)(h)# is the pyramid’s height.

Lateral Surface Area #A_L = 4 * A_t = 4 * (1/2) * 5 * sqrt(7^2 + (5/2)^2)#

#L S A = A_L = 74.3303#

Total Surface Area #color(green)( T S A = A_T = A_L + A_R = 23.097 + 74.3303 = 97.4273)#

Avery Allen · Answer

undefined To find the surface area of the pyramid, you need to calculate the areas of its base and its four triangular faces, and then sum them up.

1. **Area of the Base (Rhombus):**
The area of a rhombus can be calculated using the formula: $ \text{Area} = \frac{d_1 \times d_2}{2} $, where $ d_1 $ and $ d_2 $ are the diagonals of the rhombus.
Since the diagonals of a rhombus bisect each other at right angles, and you know the length of one diagonal (which is twice the length of one side of the rhombus, $ 2 \times 5 = 10 $), you can find the length of the other diagonal using trigonometry.
Since one corner angle of the rhombus is $ \frac{5\pi}{8} $, the adjacent angle is $ \frac{\pi}{2} - \frac{5\pi}{8} = \frac{3\pi}{8} $.
Using the cosine rule for triangles, $ \cos(\frac{3\pi}{8}) = \frac{d_2}{10} $.
Solve for $ d_2 $.
Once you have both diagonals, plug them into the area formula to find the area of the rhombus.

2. **Area of Each Triangular Face:**
The triangular faces are all identical, so you only need to calculate the area of one.
You can use Heron's formula to find the area of each triangular face.
Heron's formula states that the area ($ A $) of a triangle with side lengths $ a $, $ b $, and $ c $ is given by:
$$ A = \sqrt{s(s - a)(s - b)(s - c)} $$
where $ s $ is the semi-perimeter of the triangle, calculated as $ \frac{a + b + c}{2} $.
In this case, the side lengths are the height of the pyramid (7), and the two sides of the rhombus base (5).
Calculate the semi-perimeter and then use Heron's formula to find the area of one triangular face.

3. **Total Surface Area:**
Once you have the area of the base and the area of one triangular face, multiply the area of the triangular face by 4 (since there are four identical triangular faces), and then add the area of the base to find the total surface area of the pyramid.