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 #3 #, its base has sides of length #4 #, and its base has a corner with an angle of #(2 pi)/3 #. 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 #3  #, its base has sides of length #4 #, and its base has a corner with an angle of #(2 pi)/3   #.  What is the pyramid's surface area?

Wyatt Anderson · Accepted Answer

Total Surface Area of Pyramid #T S A = color(purple)(42.7008)#

Area of rhombic base

#A_r = a * a sin theta = 4^2 sin ((2pi)/3) = color(green)(13.8564)#

Area of slant triangle #A_s = (1/2) a * l#

where #l = sqrt((a/2)^2 + h^2) = sqrt(2^2 + 3^2) = color(brown)(3.5056)#

Lateral Surface Area of the Pyramid

#A_l = 4 * A_s = 4 * (1/2) * 4 * 3.5056 = color(green)(28.8444)#

Total Surface Area of Pyramid
#T S A = A_r + A_l = 13.8564 + 28.8444 = color(purple)(42.7008)#

Jayden Jackson · Answer

undefined The surface area of the pyramid can be calculated by finding the area of the four triangular faces and the area of the rhombus base, then summing them up.

1. Area of one triangular face:
   $$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$

2. The base of each triangle is the side of the rhombus, which is given as 4 units.
   
3. To find the height of each triangular face, we can use trigonometry. Since the angle between adjacent sides of the rhombus is $\frac{2\pi}{3}$, the height can be found using:
   $$ \text{height} = \text{base} \times \sin\left(\frac{2\pi}{3}\right) $$

4. Area of one triangular face is then $ \frac{1}{2} \times 4 \times \left(4 \times \sin\left(\frac{2\pi}{3}\right)\right) $.

5. The area of the rhombus base is $ \text{side}^2 \times \sin(\theta) $, where $ \theta $ is the acute angle between the diagonals.
   
6. Since the diagonals of a rhombus bisect each other at right angles, the diagonals of the rhombus base can be found using trigonometry:
   $$ \text{diagonal} = 2 \times \text{side} \times \sin\left(\frac{\theta}{2}\right) $$

7. Given that $ \theta = \frac{2\pi}{3} $, each diagonal is $ 2 \times 4 \times \sin\left(\frac{\frac{2\pi}{3}}{2}\right) $.

8. The area of the rhombus base is then $ 4^2 \times \sin\left(\frac{2\pi}{3}\right) $.

9. The total surface area of the pyramid is the sum of the areas of the four triangular faces and the area of the rhombus base.

10. Calculate each part and sum them up to find the total surface area of the pyramid.