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preface to fifteenth edition

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2.112 SECTION 2<br />

A h[0.4(y y ) 1.1(y y ) y y ··· y ] (Durand’s Rule)<br />

D 0 n 1 n1 2 3 n2<br />

A 1<br />

S ⁄3h[(y0 y n) 4(y1 y3 ··· y n1) 2(y2 y4 ··· y n2)]<br />

(n even, Simpson’s Rule)<br />

In general, A S gives the most accurate approximation.<br />

The greater the value of n, the greater the accuracy of approximation.<br />

Cube. V a<br />

3; d a <strong>to</strong>tal surface<br />

2<br />

p2; area 6a , where a is lengthof side and d is lengthof<br />

diagonal.<br />

2 2 2<br />

Rectangular Parallelopiped. V abc; d pa b c ; <strong>to</strong>tal surface area 2(ab bc <br />

ca), where a, b, and c are the lengths of the sides and d is lengthof diagonal.<br />

Prism or Cylinder<br />

Pyramid or Cone<br />

V (area of base) · (altitude)<br />

Lateral area (perimeter of right section) · (lateral edge)<br />

V 1⁄3(area of base) · (altitude)<br />

Lateral area of regular pyramid 1⁄2(perimeter of base) · (slant height)<br />

Frustum of Pyramid or Cone.<br />

A 2 are the areas of the bases.<br />

V 1⁄3(A 1 A2 p 1 2<br />

A · A )h, where h is the altitude and A 1 and<br />

Prismoid<br />

Lateral area of a regular figure 1⁄2(sum of perimeters of base) · (slant height)<br />

h<br />

V (A1 A2 4A 3)<br />

6<br />

where h altitude, A 1 and A 2 are the areas of the bases, and A 3 is the area of the midsection parallel<br />

<strong>to</strong> bases.<br />

Area of Surface and Volume of Regular Polyhedra of Edge l<br />

Name Type of surface Area of surface Volume<br />

Tetrahedron 4 equilateral triangles 1.73205l 2 0.11785l 3<br />

Hexahedron (cube) 6 squares 6.00000l 2 1.00000l 3<br />

Octahedron 8 equilateral triangles 3.46410l 2 0.47140l 3<br />

Dodecahedron 12 pentagons 20.64578l 2 7.66312l 3<br />

Icosahedron 20 equilateral triangles 8.66025l 2 2.18170l 3<br />

Sphere (Fig. 2.6)<br />

2 2<br />

A (sphere) 4R D<br />

A (zone) 2Rh Dh<br />

1 1<br />

4 3 1 3<br />

V (sphere) ⁄3R ⁄8D

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