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Study Notes 024, Solid Geometry 2:
By William E. Steinman:
A cone is a solid formed by joining with straight-line segments the points of a closed curve to a point outside the plane of the curve.
The point becomes the vertex.
The joining lines are called the elements.
Here is a cone.
Imagine this to be a transparent solid figure.
This particular cone is a right circular cone. A right circular cone is generated by rotating a right triangle around one of its legs as the axis.
Notice that a cone need not be a right circular cone. The base could be any closed plane figure.
The slant height of a right circular cone is the length of one of its elements.
The lateral area of a cone is the area of its conical surface excluding the base.
A spherical surface is a curved surface wherein all points on the surface are equidistant from the center point.
A sphere is a solid bounded by a spherical surface.
Think basketball.
A great circle is any circle made by a plane passing through the center of the sphere.
A small circle is a circle made by a plane cutting the sphere, but not through its center.
A minor arc is any circular arc on the sphere whose length is less than one-half the circumference.
Notice how all of these statements are really just extensions of plane geometry.
Now we can do some arithmetic.
Did I hear someone say yippee?
We looked at prisms in SN 23.
The lateral area of a prism is the sum of the areas of the lateral surfaces.
The lateral area (S) of a right prism is the product of the base perimeter (p) and the height (h) or S = ph.
The total area (A) of any prism is the sum of the lateral area and the areas of the two bases (B).
A = S + B1 + B2.
Makes sense.
So given a right triangular prism with a right triangle base with sides equal to 3, 4, and 5 inches respectively and height = 3 inches, what is the total area?
The perimeter is the total of the three sides or 3 + 4 + 5 = 12 inches.
The height is given as 3 inches.
So the later area S = ph = 12x3 = 36 sq inches.
We know the area of any triangle is ½ the base times the lateral height.
So, the area of the base of our right triangular prism is ½ of one leg times the other leg.
In this case that is ½ x 3 x 4 = 6 sq inches.
So the total area A = S + 2B = 36 + 2x6 = 48 sq inches.
Hurrah!
The lateral area of a regular pyramid is ½ the product of the slant height (l) and the base perimeter (p) or S = ½lp
How can we know that?
The faces of a pyramid are really triangles.
From plane geometry we know that the area of a triangle is ½ the lateral height (l) times the base (b).
So, in a three sided pyramid we could add the areas of the three faces to get the total area (S).
S = ½ lb1 + ½ lb2 + ½ lb3 = ½ l(b1 + b2 + b3)
Notice that the sum of the three bases is equal to the perimeter.
So S = 1/2lp.
The total area (A) of any pyramid is the sum of the lateral area and the area of the base.
A = S + B
So let’s take a regular pyramid with a slant height of 6 inches and a square base with sides equal to 3 inches.
The perimeter p = 3 inches times 4 sides = 12 inches.
The lateral area is S = ½ lp = ½ x 6 x 12 = 36 sq inches.
The base area = 32 or 9 sq inches.
The total area A = 36 + 9 = 45 sq inches.
The lateral area (S) of a right circular cylinder is the product of the circumference (C) of the base and its altitude (h).
We can easily see that if we just remove the ends then split the cylinder and lay it flat. We will have a square wherein the base is equal to the circumference of the cylinder.
We know from plane geometry that the area of a square is equal to the base times the height.
Hence, the lateral area S = Ch
If (r) is the radius of the base then S = Ch = 2 πrh.
The total area (A) of a right circular cylinder is the product of the lateral area and the areas of the two bases.
A = S +2B = 2 πrh + 2πr2 = 2πr(h + r)
So let us imagine a perfectly symmetrical beer can with a height of 5 inches and a radius of 1.25 inches.
From our equation we can find the area S =2πr(h + r)
Plugging in our values we get S =2x3.14x1.25(5 + 1.25) = 7.85x6.25 = 49.0625
Ain’t that swell?
Think this is heavy stuff?
Wait till we get to trigonometry.