Mensuration of the Circle.
All exact relations pertaining to the mensuration of the circle involve the ratio of the circumference to the diameter. This ratio, invariably denoted by π, is constant for all circles, but it does not admit of exact arithmetical expression, being of the nature of an incommensurable number. Very early in the history of geometry it was known that the circumference and area of a circle of radius r could be expressed in the forms 2πr and πr². The exact geometrical evaluation of the second quantity, viz. πr², which, in reality, is equivalent to determining a square equal in area to a circle, engaged the attention of mathematicians for many centuries. The history of these attempts, together with modern contributions to our knowledge of the value and nature of the number π, is given below (Squaring of the Circle).
The following table gives the values of this constant and several expiessions involving it:—
| Number. | Logarithm. | Number. | Logarithm. | ||
| π | 3.1415927 | 0.4971499 | π² | 9.8696044 | 0.9942997 |
| 2π | 6.2831858 | 0.7981799 | |||
| 4π | 12.5663706 | 1.0992099 | 1 | 0.0168869 | 2.2275490 |
| ½π | 1.5707963 | 0.1961199 | 6π² | ||
| 1⁄3π | 1.0471976 | 0.0200286 | √π | 1.7724539 | 0.2485750 |
| ¼π | 0.7853982 | 1.8950899 | |||
| 1⁄6π | 0.5235988 | 1.7189986 | 3√π | 1.4645919 | 0.1657166 |
| 1⁄8π | 0.3926991 | 1.5940599 | |||
| 1⁄12π | 0.2617994 | 1.4179686 | 1 | 0.5641896 | 1.7514251 |
| 4⁄3π | 4.1887902 | 0.6220886 | √π | ||
| π | 0.0174533 | 2.2418774 | 2 | 1.1283792 | 0.0524551 |
| 180 | √π | ||||
| 1 | 0.3183099 | 1.5028501 | 1 | 0.2820948 | 1.4503951 |
| π | 2√π | ||||
| 4 | 1.2732395 | 0.1049101 | 3√(6⁄π) | 0.2820948 | 1.4503951 |
| π | |||||
| 1 | 0.0795775 | 2.9097901 | 3√(3⁄4π) | 0.6203505 | 1.7926371 |
| 4π | |||||
| 180 | 57.2957795 | 1.7581226 | loge π | 1.1447299 | 0.0587030 |
| π |
Useful fractional approximations are 22⁄7 and 355⁄113.
A synopsis of the leading formula connected with the circle will now be given.
1. Circle.—Data: radius = a. Circumference = 2πa. Area = πa².
2. Arc and Sector.—Data: radius = a; θ = circular measure of angle subtended at centre by arc; c = chord of arc; c2 = chord of semi-arc; c4 = chord of quarter-arc.
Exact formulae are:—Arc = aθ, where θ may be given directly, or indirectly by the relation c = 2a sin ½θ. Area of sector = ½a²θ = ½ radius × arc.