In an inclined triangular prism, the sides of the base are 5, 6 and 9 cm. The side edge is 10 cm and makes an angle of 45 degrees with the base plane. Find the volume of the prism.
According to Heron’s theorem, we determine the area of the base of the prism.
We calculate the semiperimeter of the triangle ABC: p = (AB + BC + AC) / 2 = (9 + 5 + 6) / 2 = 10 cm.
Savs = √p * (p – AB) * (p – BC) * (p – AC) = √10 * 1 * 5 * 4 = √200 = 10 * √2 cm2.
From the vertex A1, draw a perpendicular A1O to the base ABC, then triangle AA1O is rectangular, in which the angle A1AO = 45. Then AO = A1O.
2 * A1O ^ 2 = AA1 ^ 2 = 100.
A1O ^ 2 = 50.
A1O = 5 * √2 cm.
Then Vpr = Savs * A1O = 10 * √2 * 5 * √2 = 100 cm3.
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