MENSURATION OF PLANE SURFACES.

THE following rules are mostly deduced from preceding propositions:

1. To find the area of a rectangle.

Multiply two adjacent sides together. (V. 16.)

2. To find the area of a parallelogram.

Multiply one side by the perpendicular distance to the opposite side. (V. 16, Cor. 1.)

3. To find the area of a triangle.

I. Multiply the base by one half the altitude. (V. 16, Cor. 2.) II. Multiply together the half sum of the three sides, and the excess of the half sum over the three sides respectively, and take the square root of the product.*

Hence, ABC= BC. AD = {√4a2c2 — (a2+c2 — b2)2. The quantity under the radical sign may be resolved into two factors, [2ac+a2+ c2-b2] and [2ac — (a2 + c2 — b2)], and these again into

(a+c+b)(a+c−b) and (b+a−c)(b+c−a).

Hence, the last equation becomes

4. To find the area of a trapezoid.

Multiply the sum of the parallel sides by one half the perpendicular distance between them. (V. 17.)

5. To find the area of a trapezium.

Multiply a diagonal by the half sum of the perpendiculars from the opposite angles.

6. To find the area of an irregular polygon.

I. Divide it into triangles, and find their areas separately.

II. Divide into trapezoids and triangles by a long diagonal and perpendiculars from the opposite angles, and find their areas separately. (V. 17, Sch. 1.)

7. To find the area of a regular polygon.

I. Multiply the apothem by one half the perimeter. (VI. 9.) II. Multiply the square of a side by the area of a similar polygon whose side is 1. (V. 20.)

The areas of regular polygons whose sides are unity may be calculated once for all, and tabulated.

The following table gives some of them to six places :

8. To find the circumference of a circle.

Multiply the diameter by = 3.1416, or the radius by 2π.

(VI. 11, Cor.)

9. To find the diameter from the circumference.

Divide the circumference by п.

10. To find the length of an arc of a circle.

Say, As 360° is to number of degrees in the arc, so is the whole circumference to the length of the arc. (V. 25.)

11. To find the area of a circle.

I. Multiply half the circumference by half the diameter. (VI. 12.)

II. Multiply the square of the radius by π. (VI. 12, Cor. 1.)

12. To find the area of a sector of a circle.

I. Say, As four right angles is to the angle of the sector—or, As the whole circumference is to the arc of the sector-so is the area of the circle to the area of the sector. (V. 25.)

II. Multiply the arc of the sector by the radius, and divide by two. (Proved as VI. 12.)

13. To find the area of a segment of a circle.

Subtract the area of the triangle ADC from the area of the sector ADCB.

14. To find the area of a zone* of a circle.

Subtract the area of the segment CED from the area of the segment AEB.

15. To find the area of a circular ring.

Subtract the area of the smaller circle from the area of the larger.

If R, R' be the radii, then the area of the ring is (R2 – R′2).

B

D

E

B

O

* Definition.—A zone is the portion of a circle between two parallel

chords.

13*

EXAMPLES.

1. The adjacent sides AB and AC of a parallelogram are 20 and 16, the distance on AB to the foot of the perpendicular from C is 5; what is the area? Ans. 303.8+.

2. The area of a triangle is 40 and the base is 10; what is the altitude? Ans. 8.

3. The sides of a rectangle are 16 and 25; what is the side of a square containing an equal area? of an equilateral triangle? Ans. 20; 30.39. If a be a side of an equilateral triangle, then area is equal to

a2

4

√3.

4. What would it cost to fence the above in the three cases at 50 cents per linear foot? Ans. $41; $40; $45.58.

5. A side of a regular hexagon is 10; what is the area? Ans. 259.8.

6. The gable end of a house is 40 feet wide, the height to the eaves is 40 feet, and to the crown 50 feet; how many bricks, the face of each being 8 in. by 2 in., will be required to build it, the wall being 2 bricks thick? Ans. 32,400.

7. The short sides of a right-angled triangle are 25 and 36; what is the other side? what the side of an equivalent square?

Ans. 43.8+; 21+.

8. The parallel sides of a trapezoid are 20 and 16, and the perpendicular distance between them is 6; if the other sides be produced to meet, what is the area of the whole triangle formed?

Ans. 300.

9. In the last example the trapezoid is bisected by a line parallel to the parallel sides; how far from the longer side must it be drawn?

Ans. 2.84+.

It will assist in solving this to remember that similar triangles are to each other as the squares of their homologous sides.

10. A man has three lots, each containing 10 acres ; one is a square, one a circle, and one a semicircle; which requires the least fencing to enclose it? Ans. The circle.

11. A farmer desiring to know the contents of a quadrilateral field, steps off a diagonal, 200 steps; he then goes to the other corners, and steps perpendiculars to this diagonal, 80 and 30 steps. If each step be 3 feet, how many acres in the field? Ans. 2 A. 1 R. 3 + r.

12. In a field AD was found to be 200 chains, Bb 20, Cc 18, Ee 40, Gg 10, Ff 42; also Af was 15, fb 10,

bg 15, gc 100, ce 40, and eD 20. What is the area?

Ans. 969 A.

It will be noted that the area is

ABb+ Bbc C+CcD+ Eed + EefF

+ AGg-FfgG.

C

D

E

B

A

13. One side of a regular polygon is 25, and an angle is 108°; what is the area of the polygon?

Ans. 1075.298.

14. If the diameter of the earth be 7912 miles, what is the circumference? Ans. 24856.28.

= 3.14159.

To obtain this answer use π = 15. If 15° of the equator pass under the sun in one hour, how many miles will pass in 2 hours and 22 minutes? Ans. 2451.097 miles.

16. What is the length of one degree of the equator?

Ans. 69.045 miles. 17. A circular plot 80 feet in diameter has a walk 8 feet wide running around it, inside the boundary; what amount of land is taken up by it? Ans. 1809.56.

18. A chord of a circle is 80, and its distance from the centre is 30; what is the area of the circle? Ans. 7854.

19. What is the side of a square equivalent to a circle 60 feet in diameter? Ans. 53.17. 20. A rectangle is 10 by 7; find the radius of a circle of the same perimeter. Ans. 5.41. 21. The hypotenuse of a right-angled triangle inscribed in a circle is 40; what is the area of the circle? Ans. 400π.

22. From a point without a circle a line of 8 feet is drawn touching the circle, and a line of 4 feet meeting the circle at its nearest point; what is the area of the circle? Аns. 36π.

23. What is the area of a sector of 25° in a circle whose radius is 44? Ans. 422.367.

24. What is the area of one of the segments cut off by a side of an inscribed square in a circle whose radius is 10? Ans. 28.54.

. 25. Two chords on the same side of the centre of a circle subtend angles of 60° and 40°; the radius is 14 feet; what is the area of the zone? Ans. 11.361 sq. ft.

Note. The chord f 40° to radius 1 is .684.