Answer:
The slant height of the pyramid refers to the height of one face-triangle. Now, the cube has dimensions of 2 x 2 x 2, which means the height, base and length of the pyramid are equal, that's why is a squared pyramid.
Notice that the slant height forms a right triangle with two faces of the cube, and the pyramid intercepts the cube at the middlepoint of the plane.
First, we need to find the half-length of a diagonal, which is a leg of the right triangle formed by the slant height.
[tex]d^{2}=2^{2} +2^{2}\\ d=\sqrt{4+4}= \sqrt{8}\\ d=2\sqrt{2}[/tex]
The part related with the slant height is
[tex]d_{half} =\frac{2\sqrt{2} }{2}=\sqrt{2}[/tex]
So, the slant height is
[tex]h_{slant}= \sqrt{(\sqrt{2} )^{2} +2^{2} }=\sqrt{2+4}=\sqrt{6}[/tex]
Therefore, the slant height is the square root of 6 yards.
The surface area of a squared pyramid is
[tex]S_{area}=2(2)(\sqrt{6})+(2)^{2} =4\sqrt{6}+4 \approx 13.8yd^{2}[/tex]
The five faces of the cube have a surface area
[tex]S_{cube}=5(2)^{2} =20yd^{2}[/tex]
Therefore, the composite surface area of the figure is
[tex]S_{total}=13.8+20=33.8yd^{2}[/tex]
