Rectangular Parallelepiped Problems, 17

Category: Solid Geometry

"Published in Newark, California, USA"

Find the angles that the diagonal of a rectangular parallelepiped 1 in. by 3 in. by 5 in. make with the faces.

Solution:

To illustrate the problem, it is better to draw the figure as follows 

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Next, we need to draw a diagonal line at the top and bottom of the rectangular parallelepiped as well as its diagonal and angles as follows 

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By using Pythagorean Theorem, the length of d is 

Therefore, the value of an angle is

                           or

Next, we need to draw a diagonal line at the left and right of the rectangular parallelepiped as well as its diagonal and angles as follows 

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Therefore, the value of an angle is 

                          or

Next, we need to draw a diagonal line at the front and back of the rectangular parallelepiped as well as its diagonal and angles as follows

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Therefore, the value of an angle is

                           or

Trapezoid Prism Problems, 8

Category: Solid Geometry

"Published in Newark, California, USA"

The figure shows a right section of a railroad cut in a hillside. If the sides rise 1 unit vertically to a horizontal distance of 1.5 units, and if the length of the cut is 100 ft., find the volume of earth removed.

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Solution:

The cross section of a railroad cut in a hillside is a general quadrilateral. If you draw vertical lines from the top ends of a general quadrilateral to the bottom horizontal line that contains one side, then it becomes a trapezoid as follows

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By similar triangles, we can solve for the values of a and b as follows

The area of a trapezoid is

The area of small triangle is

The area of big triangle is

Hence, the area of the base which is the cross section of a railroad cut in a hillside is

Therefore, the volume of earth removed which is the volume of a prism is

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