A pyramid has a parallelogram shaped base and a peak directly above its center. Its base's sides have lengths of #6 # and #4 # and the pyramid's height is #3 #. If one of the base's corners has an angle of #pi/4#, what is the pyramid's surface area?

Question

A pyramid has a parallelogram shaped base and a peak directly above its center.  Its base's sides have lengths of #6  # and #4  # and the pyramid's height is #3 #.  If one of the base's corners has an angle of #pi/4#, what is the pyramid's surface area?

Sadie Allen · Accepted Answer

#51.565\ ext{unit}^2# Area of parallelogram base with sides #6# & #4# & an interior angle #{\pi}/4# #=6\cdot 4\sin({\pi}/4)=16.971# The pyramid's semi-diagonal base is shaped like a parallelogram. #1/2\sqrt{6^2+4^2-2\cdot 6\cdot 4\cos({\pi}/4)}=2.125# & #1/2\sqrt{6^2+4^2-2\cdot 6\cdot 4\cos({3\pi}/4)}=4.635.# The sides of the pyramid's triangular lateral face are now listed as #\sqrt{3^2+(2.125)^2}=3.676# & #\sqrt{3^2+(4.635)^2}=5.521# There are two pairs of opposite identical triangular lateral faces of pyramid. One pair of two opposite triangular faces has the sides #6, 3.676# & #5.521# and another pair of two opposite triangular faces has the sides #4, 3.676# & #5.521# 1) Area of each of two identical triangular lateral faces with sides #6, 3.676# & #5.521# semi-perimeter of triangle, #s={6+ 3.676+5.521 }/2=7.5985# And now, the area of the pyramid's lateral triangular face using Heron's formula #=\sqrt{7.5985(7.5985-6)(7.5985-3.676)(7.5985-5.521)}# #=9.949# 2) Area of each of two identical triangular lateral faces with sides #4, 3.676# & #5.521# semi-perimeter of triangle, #s={4+ 3.676+5.521 }/2=6.5985# And now, the area of the pyramid's lateral triangular face using Heron's formula #=\sqrt{6.5985(6.5985-4)(6.5985-3.676)(6.5985-5.521)}# #=7.348# Therefore, the pyramid's total surface area (including the base area) #=2( ext{area of lateral triangular face of type-1})+2( ext{area of lateral triangular face of type-2})+ ext{area of parallelogram base}# #=2(9.949)+2(7.348)+16.971# #=51.565\ ext{unit}^2#

Evelyn Arboleda · Answer

undefined The surface area of the pyramid can be calculated by finding the areas of its different faces and then summing them up.

1. The area of the base (a parallelogram) can be calculated using the formula for the area of a parallelogram: $ \text{Area}_{\text{base}} = \text{base} \times \text{height} = 6 \times 4 $.

2. The area of each triangular face can be calculated using the formula for the area of a triangle: $ \text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} $. Since the height of the pyramid is given as 3, and the length of the base of the triangle is half of the corresponding side of the base parallelogram, the lengths of the base and height of each triangular face can be determined using trigonometric relations.

3. There are four triangular faces in total.

4. The area of each triangular face with the base's corner angle of $ \frac{\pi}{4} $ can be calculated using trigonometric relations.

5. Sum up the areas of the base and the triangular faces to find the total surface area of the pyramid.

By following these steps, the surface area of the pyramid can be determined.