Mensuration of Geometrical Figures I

These formulae are due to Archimedes.

(v.) The wedge, in the most general sense, is formed by cutting a triangular prism by any two planes (not intersecting within the prism). Its volume is = (mean of the 3 parallel edges) X (area of cross-section).

3. Moments and Centroid.

The ideas of moment and of centroid (centre of gravity) are extended to plane figures, surfaces and solids. Let F be a plane figure, L any straight line in its plane. Suppose F to be divided up into a large number n of equal elements. Let be the sum of the products obtained by multiplying the area of each element by its distance from L.

Then the limit of when n is made indefinitely great is called the moment, or first moment, of F with regard to L. The centroid of F is a point G in F (or fixed with regard to F) such that, wher ever L may be, this moment is equal to the product of the whole area of F by the distance of G from L. The centroid of a surface in general or of a solid is defined in the same way, moments being taken with regard to planes. The proof of the existence of a centroid is the same as the proof of the existence of the centroid of a material body. (See MECHANICS.) Moments of higher order than the first, i.e., second moments, third moments, • • • are ob tained by multiplying the elements by the squares, cubes, • • • of their distances from the line or plane.

4. Moments About Central Line or Plane.

When we have found moments of a plane figure with regard to a line in the plane, or of a surface or a solid with regard to a plane, we may require to find the moments about a line or plane through the centroid. Taking the case of a plane figure, let the area be and let the moments as found be • , and let x be the distance of the centroid from the original line. Then

the first moment with regard to the central line is MI' — this is o, so that x = The qth moment with regard to the The formula is the same for the moment of a surface or a volume with respect to a plane.

5. Solids and Surfaces of Revolution.—The solid or sur face generated by the revolution of a plane closed figure or a plane continuous line about a straight line in its plane, not inter secting it, is a solid of revolution or surface of revolution, the straight line being its axis. The revolution need not be complete, but may be through any angle.

The two important theorems are :— (i.) If any plane figure revolves about an external axis in its plane, the volume of the solid generated by the revolution is equal to the product of the area of the figure and the distance travelled by the centroid of the figure.

(ii.) If any line in a plane revolves about an external axis in the plane, the area of the curved surface generated by the revolution is equal to the product of the length of the line and the distance travelled by the centroid of the line.

These theorems were discovered by Pappus of Alexandria (c. A.D. 300, but possibly as early as the first century), and were made generally known by Guldinus (c. A.D. 164o). They are sometimes known as Guldinus's Theorems, but are more properly described as the Theorems of Pappas. The theorems are of use, not only for finding the volumes or areas of solids or surfaces of revolution, but also, conversely, for finding centroids or centres of gravity. They may be applied, for instance, to finding the centroid of a semi-circle or of the arc of a semi-circle.