Mensuration of Surfaces.

PROBLEM XVIII.

27. Having given the diameter or circumference of a circle, to find the side of the inscribed square.

RULE.

1st. The diameter x,7071-side of the inscribed

2d.

square.

The circumference x,2251=side of the in

2. The diameter of a circle is 412 feet: what is the side of the inscribed square.

Ans. 291,3252 sq. ft.

3. If the diameter of a circle be 600, what is the side of the inscribed square?

Ans. 424,26.

QUEST.-27. Having given the diameter of a circle, how will you find the side of an inscribed square? Having given the circumference of a cir. cle, how will you find the side of an inscribed square?

Mensuration of Surfaces.

4. The circumference of a circle is 312 feet: what is

the side of the inscribed square?

Ans. 70,2312 ft.

5. The circumference of a circle is 819 yards: what is the side of the inscribed square?

Ans. 184,3569 yds.

6. The circumference of a circle is 715: what is the

side of the inscribed square?

Ans. 160,9465.

PROBLEM XIX.

28. To find the area of a circular sector.

RULE.

1st. Find the length of the arc by Problem XII. 2d. Multiply the arc by one half the radius, and the product will be the area.

EXAMPLES.

1. What is the area of the circular sector ACB, the arc AB containing 18°, and the radius CA being equal to 3 feet.

First, ,01745 × 18 × 3 =,94230= length AB.

Then, ,94230 × 11,41345-area.

QUEST.-28. How do you find the area of a circular sector?

B

Mensuration of Surfaces.

2. What is the area of a sector of a circle in which the radius is 20 and the arc one of 22 degrees?

Ans. 76,7800.

3. Required the area of a sector whose radius is 25 and the arc of 147° 29'.

Ans. 804,2448.

4. Required the area of a semicircle in which the radius is 13.

Ans. 265,4143.

5. What is the area of a circular sector when the length of the arc is 650 feet and the radius 325?

Ans. 105625 sq. ft.

PROBLEM XX.

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

RULE.

1st. Find the area of the sector having the same arc with the segment by the last Problem.

2d. Find the area of the triangle formed by the chord

of the segment and the two radii through its extremities.

3d. If the segment is greater than the semicircle, add the two areas together; but if it is less, subtract them and the result in either case will be the area required.

QUEST.-29. How do you find the area of the segment of a circle?

And, AD=√AP2+PD2= √144+16=12,64911:

sector

but

Then, area ACB=

AGFBC=24×11,64=279,36 :

CP FP-AC-17,17-11,64-5,53:

ABX CP 20,5 × 5,53

= 56,6825.

Mensuration of Surfaces.

Then, area of sector AFBC=279,36

do. of triangle ABC= 56,6825

gives area of segment AFB-336,0425

3. What is the area of a segment, the radius of the circle being 10, and the chord of the arc 12 yards? Ans. 16,324 sq.yds.

4. Required the area of the segment of a circle whose chord is 16, and the diameter of the circle 20?

Ans. 44,5903.

5. What is the area of a segment whose arc is a quadrant-the diameter of the circle being 18?

Ans. 63,6174.

6. The diameter of a circle is 100, and the chord of the segment 60: what is the area of the segment?

PROBLEM XXI.

Ans. 408, nearly.

30. To find the area of a circular ring: that is, the area included between the circumferences of two circles, having a common centre.

QUEST.-30. How do you find the area of a circular ring?