Mensuration of Surfaces.

6. The length of an irregular figure being 37,6, and the breadths at nine equidistant places, 0; 4,4; 6,5;

7,6; 5,4; 8; 5,2; 6,5; and 6,1: what is the area? Ans. 219,255.

PROBLEM X.

18. To find the circumference of a circle when the diameter is known.

RULE.

Multiply the diameter by 3,1416, and the product will be the circumference.

EXAMPLES.

1. What is the circumference of a circle whose diame

2. What is the circumference of a circle whose diameter is 40 feet?

Ans. 125,664 feet.

QUEST.-18. How do you find the circumference of a circle when the diameter is known?

Mensuration of Surfaces.

3. What is the circumference of a circle whose diameter is 12 feet?

Ans. 37,6992 feet.

4. What is the circumference of a circle whose diameter is 22 yards?

Ans. 69,1152 yards.

5. What is the circumference of the earth-the mean diameter being about 7921 miles?

Ans. 24884,6136 miles.

PROBLEM XI.

19. To find the diameter of a circle when the circumference is known.

RULE.

Divide the circumference by the number 3,1416, and the quotient will be the diameter.

EXAMPLES.

1. The circumference of a circle is 69,1152 yards: what is the diameter ?

QUEST.-19. How do you find the diameter of a circle when the circum

ference is known?

Mensuration of Surfaces.

2. What is the diameter of a circle whose circumference is 11652,1944 feet?

Ans. 3709. ft.

3. What is the diameter of a circle whose circumference is 6850 ?

Ans. 2180,4176.

4. What is the diameter of a circle whose circumference is 50?

Ans. 15,915.

5. If the circumference of a circle is 25000,8528; what is the diameter ?

Ans. 7958.

PROBLEM XII.

20. To find the length of a circular arc, when the number of degrees which it contains, and the radius of the circle are known.

RULE.

Multiply the number of degrees by the decimal ,01745, and the product arising by the radius of the circle.

EXAMPLES.

1. What is the length of an arc of 30 degrees, in a circle whose radius is 9 feet.

QUEST.-20. How do you find the length of an arc when you know the number of degrees and the radius of the circle?

Mensuration of Surfaces.

We merely multiply the

given decimal by the number of degrees, and by the radius.

Operation.

,01745 × 30×9=4,7115, X

which is the length of the required arc.

21. REMARK. When the arc contains degrees and minutes, reduce the minutes to the decimals of a degree, which is done by dividing them by 60.

2. What is the length of an arc containing 12° 10' or 1210, the diameter of the circle being 20 yards?

Ans. 2,1231.

3. What is the length of an arc of 10° 15′ or 101°, in a circle whose diameter is 68.

Ans. 6,0813.

PROBLEM XIII.

22. To find the length of the arc of a circle when the chord and radius are given.

RULE.

1st.

Find the chord of half the arc.

2d. From eight times the chord of half the arc, sub

tract the chord of the whole arc, and divide the remainder by three, and the quotient will be the length of the arc, nearly.

QUEST.-21. If the arc contains minutes, what do you do? 22. How do you find the length of the arc of a circle when diameter and chord of the arc are known?

Mensuration of Surfaces.

EXAMPLES.

1. The chord AB=30 feet, and the radius AC=20 feet: what is the length of the arc ADB.

First, draw CD perpendicular to the chord AB: it will bisect the chord at P, and the arc of the chord at D. Then AP-15 feet. Hence.

Then,

Again,

hence,

Then,

AC2-AP2=CP2: that is

B

D

400-225-175 and √175=13,228=CP. CD-CP-20-13,22S-6,772=DP.

AD=√AP2+PD2=√225+45,859984 :

AD=16,4578-chord of the half arc.

16,4578 x 8-30

3

-33,8874-arc ADB.

2. What is the length of an arc the chord of which is 24 feet and the radius of the circle 20 feet?

Ans. 25,7309 ft.

3. The chord of an arc is 16 and the diameter of the circle 20 what is the length of the arc ?

Ans. 18,5178.

4. The chord of an arc is 50, and the chord of half the arc is 27: what is the length of the arc ?

Ans. 55.