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 has sides of length #3 #, and its base has a corner with an angle of #(3 pi)/4 #. What is the pyramid's surface area?
1 Answer
The total surface area is (approximately)
Explanation:
The pyramid is composed of 5 pieces: 1 base (a rhombus) and 4 sides (congruent triangles). (They're congruent because each triangle has a "base" of length 3, and one side is from the tip to a wide corner, while the other side is from the tip to a narrow corner.)
The surface area of the whole pyramid is
#A_"pyramid"= A_"rhombus" + 4A_"triangle"#
Step 1: The Rhombus
The area of a rhombus is
#A_"rhombus" = 3sin(pi/4) = 3/sqrt2#
Step 2: The (4) Triangles
The area of a triangle is
Then use Pythagorean theorem to find the "slant" height of the triangle side:
#a^2+b^2=c^2#
#(3/2)^2+7^2 = c^2#
#9/4+49 = c^2#
#205/4 = c^2#
#sqrt205/2 = c#
Then:
#A_"triangle" = 1/2(3)(sqrt205/2)=(3sqrt205)/4#
Step 3: Add them together!
#A_"pyramid"= A_"rhombus" + 4A_"triangle"#
#color(white)(A_"pyramid")= (3sqrt2)/2+4((3sqrt205)/4)#
#color(white)(A_"pyramid")= (3sqrt2)/2+3sqrt205#
#color(white)(A_"pyramid")~~ 45.075#