Areas Related to Circles

Before slicing anything, recall the whole. The distance once around a circle of radius r is its circumference, 2 pi r. The space inside it is its area, pi r squared. This chapter never invents new ideas about the whole circle; instead it asks: if I take only a part of the circle, what fraction of the circumference or the area do I get? The answer always comes from comparing the central angle of the part to the full 360 degrees of the whole turn.

The region enclosed by two radii and the arc between them is a sector. If the two radii make an angle theta at the centre, then theta out of 360 degrees is exactly the fraction of the whole circle you have cut off. So the arc length is that fraction of the circumference, (theta/360) x 2 pi r, and the sector area is that same fraction of the whole area, (theta/360) x pi r squared. A sector with the smaller angle is the minor sector; the rest is the major sector.

If instead of two radii you cut the circle with a straight chord, the chord and its arc enclose a segment. To find a minor segment's area, picture the sector that shares the same arc; the chord splits off a triangle formed by the two radii. The minor segment is what is left when you remove that triangle from the sector, so its area = area of sector minus area of the triangle. For a major segment you add the triangle instead, or simply subtract the minor segment from the whole circle.

Designs on tiles, brooches, flower beds and windows combine circles or their parts with squares, rectangles and triangles. The strategy is always the same: break the figure into standard pieces whose areas you know, then add the pieces you want and subtract the pieces you do not. A shaded region is found by computing the larger area and removing the unshaded parts. Careful labelling of each piece, and keeping pi as 22/7 throughout, keeps the arithmetic clean.