Area of a CircleFigure 1-11.-Trapezium.Figure 1-12.-Area of a circle.Stated in words,equal to one-halfaltitude.the area of a trapezoid isthe sum of its bases times itsArea by Reducing to TrianglesFigure 1-11 shows you how you can determinethe area of a trapezium, or of any polygon,by reducing to triangles. The dotted lineconnecting A and C divides the figure into thetriangles ABC and ACD. The area of thetrapezium obviously equals the sum of the areasof these triangles.Figure 1-12 shows how you could cut adisk into 12 equal sectors. Each of thesesectors would constitute a triangle, except forthe slight curvature of the side that was originallya segment of the circumference of the disk. Ifthis side is considered the base, then the altitudefor each triangle equals the radius (r) of theoriginal disk. The area of each triangle, then,equalsand the area of the original disk equals the sumof the areas of all the triangles. The sum of theareas of all the triangles, however, equals the sumof all the b’s, multiplied by r and divided by 2.But the sum of all the b’s equals thecircumference (c) of the original disk. Therefore,the formula for the area of a circle can beexpressed asHowever, the circumference of a circle equals theproduct of the diameter times n (Greek letter,pronounced “pi”). n is equal to 3.14159. . . Thediameter equals twice the radius; therefore, thecircumference equals 2rrr. Substituting 2rrr forc in the formulaThis is the most commonly used formula for thearea of a circle. If we find the area of the circlein terms of circumference.Area of a Segment and a SectorA segment is a part of a circle bounded by achord and its arc, as shown in figure 1-13. Theformula for its area iswhere r = the radius and n = the central anglein degrees.1-12
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