7. Area of a Trapezoid. A trapezoid is a quadrilateral having two sides parallel and two

8. Area of a Trapezium. A trapezium is a quadrilateral

9. Area of a Regular Polygon. A regular polygon is one which has equal sides and equal angles.

The apothem is the perpendicular distance from the center to any side.

The area of a regular polygon equals one-half the perimeter times the apothem..

FIG. 8.

FIG. 9.

An angle

10. Angle of a Regular Polygon. 0, of a regular polygon having n sides, equals 180° multiplied by n-2 and divided by n.

G

FIG. 13.

(

14. Area of a Ring. The area of a ring included between the circumferences of two concentric circles equals π times the difference between the squares of the two radii.

In the remaining problems in this paragraph, do not solve for the literal quantities in the second member, but do as specified in each problem.

15. Length of Arc of Sector of a Circle. A sector of a circle is a portion of it bounded by two radii and the intercepted arc.

The length of the arc of a sector of 0

0° equals times the circumference of

the circle.

360

In the formula substitute for circum

ference in terms of radius, and simplify.

FIG. 14.

16. Area of a Sector of a Circle. The area of a sector of a

circle equals one-half its radius times its arc.

In the formula, substitute for arc from the second formula of problem 15.

FIG. 15.

-a

17. Area of an Ellipse. The area of an ellipse equals times the product of the two semi-axes.

In the formula denote the semi-axes by a and b, a being half the long axis and b half the short axis.

18. Perimeter of an Ellipse. The perimeter of an ellipse equals approximately 1.82 times the long axis, plus 1.315 times the short axis.

19. Total Area of a Cylinder. The total area or surface of a cylinder, equals its circumference times its length, plus twice the area of the base.

In the formula, substitute for circumference and area of base in terms of radius of the base. Formulate, also, the curved surface of a cylinder.

FIG. 16.

20. Volume of a Cylinder. The volume of a cylinder equals the area of the base times the length of the cylinder.

In the formula substitute for area of the base in terms of radius; also in terms of diameter.

FIG. 17.

21. Total Area of a Regular Pyramid. A regular pyramid is one whose apex is directly over the center of its base which is a regular polygon.

The total area of a regular pyramid equals the area of the base Aь, plus the area of the sides AL.

In the formula substitute for A, from problem 9, and for A, from problem 5.

22. Volume of a Regular Pyramid. The volume of a regular pyramid equals one-third the area of the base times the altitude. In the formula substitute from problem 9.

23. Total Area of a Cone. The total surface or area of a cone equals the area of the base, plus the lateral area.

In the formula substitute for area of the base, in terms of radius of the base, and for lateral area substitute one-half the circumference of the base times the slant height.

In the formula thus obtained, substitute

for circumference in terms of radius.

FIG. 18.

24. Volume of a Cone. The volume of a cone equals onethird the area of the base times the altitude.

In the formula substitute for area of base in terms of radius.

25. Volumes of the Frustum of a Cone and a Pyramid. A frustum of a cone and of a pyramid is the part which remains when the top is cut off parallel to the base.

FIG. 19.

The volume of a frustum of a cone and of a pyramid equals one-third the altitude times the following:

Area lower base AB, plus area upper base Aь, plus the square root of the product of the areas of the two bases.

FIG. 20.

26. Area of a Sphere. The area of a sphere equals π times the square of its diameter.

Formulate also in terms of radius.

27. Volume of a Sphere. The volume of a sphere equals two-thirds π times the square of the radius, times the diameter.

In the formula substitute for radius in terms of diameter, and simplify; also

substitute for diameter in terms of radius, and simplify.

28. Volume of a Rectangu

lar Solid. Write a law for the volume of the solid here shown, and formulate the law.

20

FIG. 21.

h

29. Volume of a Cylindrical Ring. The volume of a ring with a circular cross-section, equals 2.4674 times the square of the thickness, times the sum of the thickness and the inner diameter.