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Development of Formula:
The above demonstrates that a circle divided into a number of parts maybe laid out as a parallelogram. As the number of parts is increased, the longer side approaches 1/2 of the circumference; if divided into an indefinite number of parts, it would be equal to 1/2 the circumference. As the number increases, h approaches r and would be equal if the circle was divided into an infinite number of parts. The areas of the parallelogram is
The formula for the area of a circle
Since r = d/2 where d is the diameter of a circle, the formula for the area of a circle in terms of its diameter is
GEOMETRIC SOLIDS In describing plane shapes, you use only two dimensions: width and length; there is no thickness. By adding the third dimension, you describe a solid object. Consider the solids shown below.
1. A PRISM is a figure whose two bases are polygons, alike in size and shape, lying in parallel planes and whose lateral edges connect corresponding vertices and are parallel and equal in length. A prism is a right prism if the lateral edge is perpendicular the base. The altitude of a prism is the perpendicular distance between the bases. 2. A CONE is a figure generated by a line moving in such a manner that one end stays fixed at a point called the "vertex." The line constantly touches a plane curve which is the base of the cone. A cone is a circular cone if its base is a circle. A circular cone is a right circular cone if the line generating it is constant in length. The altitude of a cone is the length of a perpendicular to the plane of the base drawn from the vertex. 3. A PYRAMID is a figure whose base is a plane shape bounded by straight lines and whose sides are triangular plane shapes connecting the vertex and a line of the base. A regular pyramid is one whose base is a regular polygon and whose vertex lies on a perpendicular to the base at its center. The altitude of a pyramid is the length of a perpendicular to the plane of the base drawn from the vertex. 4. A CIRCULAR CYLINDER is a figure whose bases are circles lying in parallel planes connected by a curved lateral surface. A right circular cylinder is one whose lateral surface is perpendicular to the base. (Note: Any reference in this text to a cylinder will mean a circular cylinder.) The altitude of a circular cylinder is the perpendicular distance between the planes of the two bases. |
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