The figure above shows a solid formed by joining the bases of two square pyramids to opposite faces of a cube. If each edge of the solid has length inches, what is the total surface area, in square inches, of this solid
(A) 100 + 25√3
(B) 100 + 50√3
(C) 150 + 25√3
(D) 200
(E) 150 + 50√3
Choice (B) is correct. The surface of the solid is made up of 4 square faces of the cube plus the triangular (lateral) faces of the pyramids. Each face of the cube has an area of,(5)(5)=25 and so the 4 faces of the cube have a total surface area of 100 square inches. Each lateral face of the pyramids is an equilateral triangle, which can be divided into two 30°,60°,90° triangles by a perpendicular dropped from any of the triangle’s vertices. Each 30°,60°,90° triangle will have a base of 2.5, which is half of the side length 5, since the hypotenuse of the triangle is 5 and the triangle is 30°,60°,90°; the second leg of each triangle is 2.5√3. Since the area of a triangle is 1/2bh, the area of each triangle is
. Each lateral face of each of the pyramids is made up of two congruent triangles, thus the area of each lateral face of the pyramids is
. There are a total of 8 equilateral triangular faces on the solid, so the total area of triangular faces is
. Therefore, the total surface area of the solid is
.