MENSURATION.

NOTE. The definitions of various lines, surfaces and solids, are given on pages 159, 162, 168. Such as are in general use, and not found there, are given in this section.

(1.)

TRIANGLES.

N

Right-angled triangle.

N

Equilateral triangle.

N Isosceles triangle.

Define a triangle; a right-angled triangle. (See Arts. 550, 551.)

565. A triangle which has its sides all equal, as Fig. 2, is an equilateral triangle.

566. A triangle which has two of its sides equal, as Fig. 3, is an isosceles triangle.

EXERCISES.

Draw a right-angled triangle; an equilateral triangle; an isosceles triangle; a right-angled isosceles triangle.

567. Each of the preceding figures has four sides and four angles; such figures are quadrilaterals.

NOTE.

-B Lines which are equally distant from each
-Dother throughout their whole extent, as A B and

C D, are parallel lines.

568. Figs. 4, 5, 6, and 7 have their opposite sides parallel; such figures are parallelograms.

569. A parallelogram whose angles are all right angles, as Figs. 6 and 7, is a rectangle.

570. A rectangle, whose sides are all equal, as Fig. 7, is a square.

571. A quadrilateral, only two of whose sides are parallel, as Fig. 8, is a trapezoid.

572. The name polygon is applied to any figure bounded by straight lines. A polygon of five sides is a pentagon, one of six sides is a hexagon, one of eight sides is an octagon, and one of ten sides is a decagon. A polygon whose sides are all equal is an equilateral polygon; a polygon whose angles are all equal is an equiangular polygon. A polygon that is equilateral

and equiangular is a regular polygon.

EXERCISES.

Draw a parallelogram; a rectangle; a square; a trapezoid; a pentagon; a regular hexagon; an octagon.

573. The line upon which a figure is supposed to stand, as O S, Fig. 4, is the base.

574. In any figure, the shortest distance from the farthest point above the base to the line of the base, as M N, Fig. 4, is the height.

575. A line joining any two angles of a figure, not adjacent, as O P, Fig. 4, is a diagonal.

EXERCISES.

What lines are bases of the preceding figures? What lines indicate heights? What are diagonals?

TO FIND THE AREAS OF RECTILINEAR FIGURES.

о в

576. If in the parallelogram A B C D the part D M C, cut off by the perpendicular M N, is placed upon the other side of the figure at A O B, a rectangle A O M D will be formed, which will have the same base and height as the parallelogram A B C D, and the same area. In the same way it Ak may be shown that the area of any parallelogram

equals that of a rectangle of the same base and height. Hence

To find the area of a RECTANGLE, a SQUARE, or any PARALLELOGRAM, Multiply the number of units in the base by the number of like units in the height. (Art. 310.) The product will equal the number of square units in the area.

B

Hence,

577. If the parallelogram A B D C is cut by the diagonal B C, the two triangles. thus formed will be found to be exactly C alike; therefore the triangle A B C equals one half of the parallelogram A B D C.

To find the area of a TRIANGLE, Multiply the number of units in the base by half the number of like units in the height.

578. The trapezoid A B M O may be cut by the diagonal A M into two triangles whose bases are the parallel sides of the trapezoid and whose height is the distance between them. Hence,

O M

To find the area of a TRAPEZOID, Multiply one half of the sum of the number of units of the two parallel sides by the number of like units in the distance between them.

579. To find the area of any POLYGON, Divide it into triangles and find the sum of their areas.

580. EXAMPLES.

NOTE.-The pupil will be greatly assisted by drawing figures to illustrate the examples which follow.

1. How many square feet are there in the surface of both sides of a slate that is 12 inches long and 8 inches wide ?

2. How many square yards in the surface of a table that is 8 feet long and 3 feet wide?

3. How many square yards in the surface of a floor that is 10 feet long and 9 feet 2 inches wide?

4. How many square rods in a garden that is 42 feet square?

Ans. 65 rds.

5. How many acres in a rectangular field that is 20 chains long and 18 chains 20 links wide? (Art. 311, Note II.)

Ans. 37.31 A.

6. What is the area of a triangle whose base is 7 feet and whose height is 2 feet 8 inches? Ans. 91 sq. ft.

7. What is the area of a right-angled triangle whose base is 3 feet and whose perpendicular is 2 feet?

8. What is the area of a right-angled triangle whose base is 5 feet and whose hypothenuse is 13 feet? (See Art. 553, Rule II.)

9. What is the area of a right-angled triangle whose perpendicular is 3 ft. 8 in. and whose hypothenuse is 4 ft. 7 in.?

M

B

N

10. What are the height and area of a triangle each of whose sides is 20 feet long?

NOTE.

- The perpendicular M N divides the equilateral triangle into two equal right-angled triangles; therefore the line A N equals of A B.

Ans. Height 17.32 ft.+; Area 173.2 sq. ft.+

11. What are the height and area of a triangle whose base is 3 ft. in length, and each of the other two sides 2 ft. 6 in. ?

Ans. Height 2 ft.; Area 3 sq. ft.

12. How many square inches in a trapezoid whose parallel sides are 5 and 6 inches long respectively, and the distance between them 4 inches ? Ans. 22 sq. in.

13. What is the area of a board 8 ft. long, one end of which is 2 ft. wide and the other 1 ft. 9 in. ?

14. I have an irregular four-sided field a diagonal of which measures 5 ch. 3 1., and lines drawn from the opposite corners perpendicular to the diagonal, 2 ch. 8 l. and 4 ch. 5 l. respectively; how many acres are there in the field? Ans. 1.54 A.+

15. What is the number of square rods in a field shaped like the figure in the margin, and whose measurements are as follows: A C 6 rd.; FAD 9 rd.; A E 10 rd.; B M 2 rd.; CN 3 rd.; DO 5 rd.; P F 4 rd.? Ans. 68 sq. rds.

CIRCLES.

Define circle; circumference; radius; diameter; arc. (See Arts. 326-329.)

581. A circle may be considered as made up of triangles whose bases form the circumference of the

circle and whose vertices are at the B

centre, the height of the triangles

being equal to the radius of the circle; hence,

A

D

The number of square units in the area of the circle is equal to the number of units in the circumference multiplied by half the number of like units in the radius, or by one fourth the number of like units in the diameter.

582. It has been found that the circumference of every circle equals its diameter multiplied by 3.1416 nearly; hence