A pyramid has a parallelogram shaped base and a peak directly above its center. Its base's sides have lengths of #6 # and #3 # and the pyramid's height is #7 #. If one of the base's corners has an angle of #(5pi)/6#, 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 #3  # and the pyramid's height is #7 #.  If one of the base's corners has an angle of #(5pi)/6#, what is the pyramid's surface area?

Aaliyah Anderson · Accepted Answer

#V_("pyr")=1/3|6*3|sin(5/6pi)*7 = 21 "units"^3#
Given : parallelogram (quadrilateral) pyramid with sides, height and angle between sides as follows - #AB=6; AC=3, EF=7# and #/_BAC=5/6pi# Required: Volume? Solution Strategy: Use the general pyramid volume formula. a) #V_("pyr") = 1/3("Base Area" xx "Height")=1/3A_("pllgm")xxH# b) #A_("pllgm") = |s_1*s_2|*sintheta# So inserting b) into a) we get the volume #V_("pyr")=1/3|bar(AB)*bar(AC)|sintheta*bar(EF) # substituting #V_("pyr")=1/3|6*3|sin(5/6pi)*7 = 21 "cubic units"#

Aaliyah Anderson · Answer

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

1. **Area of the Base:**
   Since the base is a parallelogram, its area can be calculated as the product of one of its sides and its corresponding height. Given that the base sides are 6 and 3, we can choose either side length. Let's choose 6.
   Area of the base = base side length * height = 6 * 7 = 42 square units.

2. **Area of the Triangular Faces:**
   The triangular faces of the pyramid are isosceles triangles. We need to find the lengths of the two equal sides to calculate their area.
   Using trigonometry, we can find the length of the equal sides:
   $$ \text{Side length} = \frac{{\text{Adjacent side}}}{{\cos(\theta)}} $$
   where $ \theta = \frac{5\pi}{6} $.

Given that one side of the parallelogram base is 6, and the angle opposite to it is $ \frac{5\pi}{6} $, we can find the length of the equal sides:
   $$ \text{Side length} = \frac{6}{\cos\left(\frac{5\pi}{6}\right)} $$

3. **Area of Each Triangular Face:**
   The area of an isosceles triangle can be calculated using the formula:
   $$ \text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} $$

4. **Total Surface Area:**
   Once we have the area of the base and the area of each triangular face, we can find the total surface area by summing them up.

Performing these calculations will give us the surface area of the pyramid.