Square Prism Problems, 4

Category: Solid Geometry

"Published in Vacaville, California, USA"

The figure represents a truncated prism. The base ABCD is a square, and the lateral edges AE, BF, CG, DH are perpendicular to the base. AB = 10 in., AE = 6 in., BF = 10 in., CG = 10 in., DH = 6 in. Find (a) the length of each face diagonal; (b) each face angle; (c) the length of each diagonal of solid; (d) the lateral area; (e) the total area; (f) the angle BFH; (g) the volume.

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Solution:

(a) Label further the figure and draw a diagonal line at each face as follows
 

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By Pythagorean Theorem, the length of the diagonals are

(b) Let's consider the side view of a truncated prism which is HDCG as follows

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By using basic trigonometric functions, we can solve for the face angles as follows

                            or

                            or

The opposite side of a truncated prism which is AEFB, their face angles are also 68°11'55'' and 111°48'5''. The rest of the face angles are all 90°. 

(c) Let's consider the given figure again and draw their diagonal lines as follows

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The length of the diagonals of a truncated prism are

(d) The lateral area of a truncated prism is

(e) The total area of a truncated prism is

(f) Let's consider the given figure again as follows

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By Cosine Law, the value of ∠BFH which is also α is

                             or

(g) The volume of a truncated prism is

Triangular Prism Problems, 6

Category: Solid Geometry

"Published in Vacaville, California, USA"

In the figure is shown a block of wood in the form of a right prism whose bases are right triangles. A cylindrical auger hole of diameter 2 in. is bored through the block. If the elements of the cylindrical hole are perpendicular to face ABEF and if the lateral surface of the cylindrical hole is tangent to face ABCD, find the volume removed.

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Solution:

Did you notice that the altitude or length of a right triangular prism is not given in the problem? Well, that's fine because we don't need it in the problem. If the word problem says "If the elements of the cylindrical hole are perpendicular to face ABEF and if the lateral surface of the cylindrical hole is tangent to face ABCD,..", then the side view of the section will be like this
 

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The darker section is a truncated right circular cylinder because the two bases are not equal and their altitudes or elements are not equal. We have to get the average of their altitudes first before we can solve for the volume of a cylinder. By similar triangles, the other altitude of a cylinder is

 

 

 

Therefore, the amount of a cylindrical auger hole removed from a prism which is the volume of a truncated right circular cylinder is