Question

A pyramid has a parallelogram shaped base and a peak directly above its center. Its base's sides have lengths of `2` and `8` and the pyramid's height is `2`. If one of the base's corners has an angle of `(5pi)/6`, what is the pyramid's surface area?

Answer 1

Total Surface Area # T S A = 34.8328

Explanation:

`CH = 2 * sin (5pi/6)= 1`
Area of parallelogram base `= 8 * b1 = 8 * 1 = 8`

`EF = h_1 = sqrt(h^2 + (a/2)^2) = sqrt(2^2+ (8/2)^2)= 4.4721`
Area of `Delta AED = BEC = (1/2)*b*h_1 = (1/2)*2*4.4721 = 4.4721`

`EG = h_2 = sqrt(h^2+(b/2)^2 ) = sqrt(2^2+(2/2)^2 )= 2.2361`
Area of `Delta = CED = AEC = (1/2)*a*h_2 = (1/2)*8*2.2361= 8.9443`

Lateral surface area = `2* DeltaAED + 2*Delta CED`
`=( 2 * 4.4721)+ (2* 8.9443)= 26.8328`

Total surface area =Area of parallelogram base + Lateral surface area `= 8 + 26.8328 = 34.8328`

Answer 2

The pyramid's surface area is `8+10sqrt(21+4sqrt(3))`

Explanation:

I recommend to use coordinates to solve this problem.

The pyramid's base is ABCD and its center is E

The coordinates of A and B are:
`A(0;0)`
`B(8;0)`

In order to find the coordinates of C and D, we need to use trigonometry and the pythagorean theorem:

`l=2cos(pi/6)=2sqrt(3)/2=sqrt(3)`
`L=sqrt(2^2-sqrt(3)^2)=sqrt(4-3)=sqrt(1)=1`

Therefore the coordinates of C and D are:
`C(8+sqrt(3);1)`
`D(sqrt(3);1)`

A parallelogram's diagonals cross mid-length, thus the coordinates of E are:
`E(4+sqrt(3)/2;1/2)`

If the base is located in the (x;y) plane and the pyramid's peak (called F) is located at 2 above E, then the coordinates of all the pyramid's points are:
`A(0;0;0)`
`B(8;0;0)`
`C(8+sqrt(3);1;0)`
`D(sqrt(3);1;0)`
`F(4+sqrt(3)/2;1/2;2)`

The pyramid's base's surface area is `AB*L=8*1=8`

The ADF and BCF sides' surface areas are:
`(AF*AD)/2=(sqrt((4+sqrt(3)/2)^2+(1/2)^2+2^2)*cancel2)/cancel2=sqrt((16+4sqrt(3)+3/4)+1/4+4)=sqrt(21+4sqrt(3))=AF`

The ABF and CDF sides' surface areas are:
`(AF*AB)/2=AF*4=4sqrt(21+4sqrt(3))`

Therefore, the pyramid's total surface area is:
`8+2sqrt(21+4sqrt(3))+8sqrt(21+4sqrt(3))=8+10sqrt(21+4sqrt(3))`