A Rectangle (Art. 166) is a right-angled parallelogram ; a Rhomboid is a parallelogram having no right angles; and a Rhombus is a rhomboid having equal sides.

B

E

A

F

D

394. It will be seen by the diagram that the rhombus ABCD is equal to the rectangle EBCF of the same base and altitude (Art. 218). Hence,

The area of a parallelogram is equal to the product of the base and altitude.

9. What is the area of a parallelogram whose base is 36 feet and altitude 15 feet?

10. The base of a rhombus is 16 feet and its height 12 feet. What is its area?

11. What is the difference in the area of two floors, the one being 37 feet long and 27 feet wide and the other 40 feet long and 20 feet wide?

A Trapezoid.

395. A Trapezoid is a quadrilateral having only two of its sides parallel.

396. The area of a trapezoid is equal

to the product of half the sum of the par

allel sides and the altitude.

12. What is the area of a trapezoid, the longer of the two parallel sides being 120 feet, the shorter 100 feet, and the altitude 85 feet?

13. What is the area of a plank whose length is 6 meters, the width of one of the parallel ends being 60 centimeters and the other 40 centimeters?

14. The parallel sides of a field are 131 and 243 yards, and the breadth 220 yards. How many acres does it contain?

397. A Trapezium is a quadrilateral having no two of its sides parallel.

A Diagonal is a straight line joining any two angles of a plane figure not adjacent, as the line A C.

A Trapezium.

398. It will be seen from the above diagram that a diagonal divides a trapezium into two triangles. Hence,

The area of a trapezium is equal to the product of the diagonal and half the sum of the perpendiculars drawn to the diagonal from the vertices of opposite angles.

NOTE 1. Any plane figure bounded by straight lines is called a Polygon, and may be divided into triangles; and the sum of the areas of the triangles will be the area of the figure.

NOTE 2. For the Circle see Art. 221.

15. The diagonal of a trapezium is 16 feet, and the perpendiculars upon it from the opposite angles are 7 feet and 5 feet. Find the area.

16. What is the area of a trapezium whose diagonal is 65 feet, and the length of the perpendiculars let fall upon it from opposite angles is 14 feet and 18 feet?

17. How many square yards of paving are there in a trapezium whose diagonal is found to measure 126 feet 3 inches, and the perpendiculars upon it 58 feet 6 inches and 65 feet 9 inches?

PRISMS.

399. A Prism is a body having two equal parallel polygons as bases and the other faces parallelograms.

A prism is triangular, quadrangular, pentagonal, etc., according as its bases have three sides, four sides, five sides,

A Triangular
Prism.

A Quadrangular
Prism.

etc.

400. The contents of a prism are equal to the product of the area of the base by the altitude or length.

NOTE. For the Cylinder see Art. 225.

18. What are the contents of a triangular prism whose length is 15 feet and the area of its triangular base is 6 square feet?

19. What are the contents of a quadrangular prism whose length is 6 meters, and whose base is 18 by 20 centimeters? 20. The altitude of a pentagonal prism is 20 and the area of its base 1075.30 square inches. contents in cubic feet?

feet 6 inches, What are its

A Pyramid.

PYRAMIDS AND CONES.

401. A Pyramid is a body whose base is any polygon, and whose sides are triangles meeting at a point called the vertex of the pyramid.

A pyramid, like a prism, is triangular, quadrangular, pentagonal, etc., according to the form of its base.

402. A Cone is a body whose base is a circle, and whose convex surface tapers uniformly to a point called the vertex of the cone.

The altitude of a pyramid or cone is the shortest distance from the vertex to the center of the base; as AB.

The slant height is the shortest distance from the vertex to the perimeter of the base; as A C.

403. The contents of a pyramid or cone are equal to the product of its base by one third of the altitude.

21. What are the contents of a triangular pyramid whose altitude is 14 feet 3 inches, and the area of whose base is 14.70 square feet?

22. What are the contents of a cone whose altitude is 15.06 meters, the circumference of the base being 12.5 meters?

23. The largest of the Egyptian pyramids is square at its base, and measures 693 feet on a side. Suppose its other sides to meet at a point 500 feet above the base. What are the contents of the pyramid in cubic feet?

cone are equal to the sum of the areas of the two bases plus the square root of their product, multiplied by a third of the altitude.

24. What are the contents of the frustum of a square pyramid whose altitude is 30 feet, and whose side at the base is 20 feet and at the top 10 feet?

Solution.

20 x 20 400; 10 x 10 = 100; 400 x 100 = 40000.

√ 40000 = 200; 200+ 400+ 100 700.

30310; 700 x 10 = 7000 cu. ft.

25. What are the contents of a column whose altitude is 28 feet 6 inches, and whose diameter at the larger end is 3 feet and at the other 2 feet 6 inches?

26. How many cubic feet in a square stick of timber whose length is 18 feet 8 inches, and whose side at the larger end is 27 inches and at the smaller is 16 inches?

A Sphere.

406. A Sphere is a body bounded by a curved surface, all parts of which are equally distant from a point within called the center.

The Diameter of a sphere is any straight line drawn through its center, and terminating both ways in the surface; and the Circumference is the greatest distance around the sphere.

407. The surface of a sphere is equal to the product of 3.1416 by the square of the diameter; and

The contents of a sphere are equal to the product of of 3.1416 by the cube of the diameter.

27. What is the surface of a sphere whose diameter is 25 inches?

28. What number of square meters of gold-leaf will gild a globe 18 centimeters in diameter ?