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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.

394. It will be seen by the diagram that the rhombus A B C D is equal to the rectangle EBCF of the

same base and altitude (Art. 218). E

Hence, A F D 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 ?

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

396. The area of a trapezoid is equal A Trapezoid. to the product of half the sum of the parallel 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.

7 ft. + 5 ft.
- Solution. — - " X 16 = 96 sq. ft. .

2

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,

etc.

A Triangular

Prism.

Prism.

400. The contents of a prism are equal to the product of the

area of the base by the altitude A Quadrangular

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 wbose base is 18 by 20 centimeters ?

20. The altitude of a pentagonal prism is 20 feet 6 inches, and the area of its base 1075.30 square inches. What are its contents in cubic feet?

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.

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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 A B.

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.

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A Cone.

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 ?

404. A Frustum of a pyramid or a cone is the part which remains after cutting off the top by a plane parallel to the base.

405. The contents of the Pyramid. jrustum of a pyramid or Cone. 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.

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Frustum of a

Frustum of a

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.

V 40000 = 200; 200 + 400 + 100 = 700.

30 ; 3= 10; 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 į 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 ?

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