To find the volume of this prism, we have to calculate the area of the base, B, and then multiply this number by the height of the prism, h.
V=Bh
To find the area of the base we will divide it into 6 triangles, each with a central angle of 360^(∘)6=60^(∘).
Examining the diagram, we can identify a 30^(∘)-60^(∘)-90^(∘) triangle. In such a triangle the longer leg is sqrt(3) longer than the shorter leg. With this information, we can determine the height of the triangle to 7sqrt(3).
Now we have enough information to calculate the area of one of the triangles. If we multiply this number by 6, we get the area of the prism's base.
(1/2(14)(7sqrt(3)))6=294sqrt(3)
Since we know the prism's volume and we now know the area of the base, we can find the height of the prism.
The height is 5 cm. To find the surface area, we have to determine the area of all the external faces of the solid. We already know the area of the prism's base. Since we have two of them, we will double this number to obtain the surface area of both bases.
2(294 sqrt(3))=588 sqrt(3) cm^2
We also have 6 rectangular faces with a length and width of 14 cm and 5 cm, respectively. By multiplying these dimension we get the surface area of one of these faces. If we multiply this number by 6 we get the surface area of all the rectangular faces.
(5(14))6=420 cm^2
Finally, we add all of the surfaces areas to get the total surface area.
420+588 sqrt(3) ≈ 1438 cm^2
