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MENSURATION.

In the following formulas, the letters have the meanings

here given, unless otherwise stated.

D

larger diameter;

d

smaller diameter;

R

=

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=

=

=

=

p

perimeter or circumference;

C = area of convex surface = area of flat surface which can be rolled into the shape shown;

S = area of entire surface

C+ area of the end or ends;

A

π

area of plane figure;
3.1416, nearly ratio of any circumference to its diam-
eter;

=

=

radius corresponding to D;

radius corresponding to d;

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V = volume of solid.

The other letters used will be found on the cuts.

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=

p 2√π A = 3.5449 A.

2 A 4 A

r

d

d

CIRCLE.

= πd

= 3.1416 d.

= 2πr= 6.2832 r.

=

A

=

=

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=

p

π

2V

p

2 π

=

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=

A

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πd2

= .7854 d2.

4

r2 = 3.1416 r2.

pd

=

4'

.3183 p.

.5642 VA.

.1592 p.

BB

D= B+ C.
BD-C.

E'

= E.

The above letters refer to angles.

For a right-angled triangle, c being the hypotenuse,

b

c = √ a2 + b2.

a = c2 b2.

b

V c2 a2.

C =

length of side opposite an acute angle of an oblique-angled triangle.

C = Va2 + b2

h Va2 €2.

c = length of side opposite an obtuse angle of an oblique-angled triangle.

For any triangle,

A

a

A

b

TRIANGLES.

C = Va2 + b2 + 2b e.

h = √ a2 — e2.

For a triangle inscribed in a semicircle; i. e., any rightangled triangle,

=

bh

2

Vaz

E+B+C = 180°.
E' +B+C
B' = B.

= 180°.

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2be.

=

a

a: bee: a = h: c.

c2

a

b2.

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La

b

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RECTANGLE AND PARALLELOGRAM.

A = ab.

A = by c2

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TRAPEZIUM.

Divide into two triangles and a trapezoid.

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n

+c+ call the altitudes of the triangles h and ha; then A = (h1 + h2).

A =

A = bh' + a(h' + h) + ch; or, A = [bh'+ch + a(h'+h)].

Or, divide into two triangles by drawing a diagonal. Consider the diagonal as the base of both triangles, call its length 1;

π

4

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D2 + d2

2

A =

Dd

1=

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ELLIPSE.

A = lr.

.7854 D d.

SECTOR.

π r2 E

E

(D-d)2
8.8

= .008727 2 E.

SEGMENT.

A = [lrc(r -- h)].

πγ2 Ε с

A=

2

= .0175 r E.

360
length of arc.

=

(r− h).

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360

πγΕ

180

180 /

57.2956

π r

r

*The perimeter of an ellipse cannot be exactly determined without a very elaborate calculation, and this formula is merely an approximation giving fairly close results.

D

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PRISMOID.

A prismoid is a solid having two parallel plane ends, the edges of which are connected by plane triangular or quadrilateral surfaces.

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