#1**+1 **

Every edge of SPQR has length 18 or 41, but no face of SPQR is equilateral. So every face of SPQR is a triangle with sides of length 41, 41, and 18 or 18, 18, and 41 . However, it is not possible to have a triangle with sides of length 18, 18, and 41 because 18 + 18 < 41 . So all of the faces of SPQR must be triangles with sides of length 41, 41 and 18 .

And I'm pretty sure this is possible because you can imagine folding a piece of paper in half and cutting an isosceles triangle out of the paper so that the base is the folded edge and is 18 units long, and the two legs are 41 units long. Then if you unfold the paper, you get two congruent isosceles triangles connected at the base. Unfold the paper so that the distance between the two vertex angles is 18 units. Then if you connected those two vertex angles together with a line, you would get SPQR.

The surface area of SPQR is 4 times the area of one of these faces.

area of one face = (1/2)(base)(height)

Let the base be 18 .

And from the Pythagorean theorem we can say...

height = √[ 41^{2} - 9^{2} ]

height = √[ 1681 - 81 ]

height = √[ 1600 ]

height = 40

area of one face = (1/2)(18)(40)

area of one face = 360

surface area of SPQR = 4 * 360

surface area of SPQR = 1440 sq units

hectictar Oct 11, 2018