The length, width and height of a quadrangular prism (cuboid) are 8cm, 4cm and 20cm respectively, so the surface area and volume of the quadrangular prism (cuboid) are respectively:
S quadrangular prism = (8× 4+4× 20+8× 20 )× 2 = 272× 2 = 544cm2,
V quadrangular prism = 8×4×20 = 640 cubic centimeters
Surface area and volume of quadrangular prism.
We know that the height of a quadrangular prism is 2cm, the upper bottom surface is a square with a side length of 12cm, and the lower bottom surface is a square with a side length of 20cm. As we know, the surface area of a quadrangular prism is equal to the sum of the four side areas of the quadrangular prism and the upper and lower bottom areas. Therefore, the key is to find out the areas of the four sides of a quadrangular prism. Because its four sides are equal in area, our Lord requires the area of one of them. Problem solved. Let's first find out the oblique height on the ABCD plane in the quadrangular prism, so that AE⊥CD passes through point A, and connect the vertical bottom of AO to point O. It is known that AO=2cm and AE is the oblique height on the ABCD plane in the quadrangular prism:
∴AE=20- 1222+22=25cm, so the surface area and volume of a quadrangular prism are:
S four edges =S four edges +S upper bottom +S lower bottom = 4×12+202× 25+12×12+20× 20.
=( 1285+544)cm2，
V Tetraprism =1312×12+02×12+20× 20+20× 20× 2.
=23544+434cm3。
We know that surface area is the area of geometric surface, which represents the size of geometric surface; Volume is the space occupied by geometric figures. Therefore, the surface area of sphere, quadrangular prism and quadrangular prism is not the surface area of trophy. The surface area of the trophy should be the sum of the two bottom areas of the quadrangular prism:
∴ The surface area of the trophy is S=S ball +S quadrangle +S quadrangle -2×S quadrangle bottom.
= 16π+544+ 1285+544-2×(4×8)
= 16π+ 1024+ 1285
≈ 1360cm2，
The volume of the trophy V=V ball +V quadrangular prism +V quadrangular prism =323π+640+23434+544.
≈ 1052cm3。
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