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If the segment be less than a semicircle, to find its area :

Find the area of the whole sector abcd, of which it is a part, and from the area of the sector subtract the area of the triangle acd, included between the radii and chord, and the remainder will be the area of the less segment.

To find the area of a segment greater than a semicircle:

Find the area of the whole sector, to which add the area of the triangle abc, included between the radii and the chord.

To find the arc line in the greater segment :-Divide the arc into two equal parts, and then proceed with each half as directed under the sector.

EXAMPLES.

1. What is the area of a segment less than a semicircle, whose chord is 172, the chord of half the arc 104, and the versed sine, or height of the arc, 58.48? Ans. 7248.25.

By the rules under the sector, we find the arc line, abc, to be 220, the diameter of the circle 184.95, and the area of the sector 10172.25; and by the rules under the triangle, ¶ 19, we find the altitude of the triangle to be 33.995, and its area 292, which, subtracted from the area of the sector, leaves 7248.25 for the area of the segment.

2. In a greater segment, the chord is 136, the chord of the arc is 146, the chord of of the arc is 86, and the radius of the circle is 80; what is the area of the segment?

Ans. 17309.28.

We find the arc of half the segment to be 180.666, and the whole arc 361.332, and the area of the sector 14453.28, the altitude of the triangle 42 nearly, and its area 2856, which, being added to the area of the sector, gives 17309.28 for the area of the circular segment.

3. What is the area of a circular segment less than a semicircle, its chord being 40, and height 4? Ans. 107.515.

4. The chord of a segment is 20, and its height is 5; what is its area?

Ans. 69.81.

5. What is the area of a segment, whose chord is 16, and the diameter of the circle 20 ? Ans. 44.73.

6. What is the area of a segment, whose arc is a quadrant, or 90 degrees, and the diameter 12 feet?

Ans. 10.274 square feet.

The chord in this example will evidently be equal to the side of a square inscribed in the circle; and consequently, to find the chord multiply the diameter by .7071065; and the height of the triangle will be half the length of the chord, and the length of the arc will be one-fourth the circumference of the circle.

The area may also be found by means of a table containing the areas of segments of a circle, whose diameter is 1, and whose heights, or versed sines, are all the numbers between 0 and 1 carried to any number of decimal places.

To find the area of a circular segment by this method :— Divide the versed sine by the diameter of the circle, (and the quotient is the height of the similar segment when the diameter is 1) take the tubular area corresponding to the quotient, and multiply it by the square of the diameter, and the product is the area of the given segment.

EXAMPLES.

1. The diameter of a circle is 30 feet, and the versed sine of the segment is 6 feet; what is its area? Ans. 100.64 feet.

Solution.-.2, the height of the similar segment whose diameter is 1. Look for .2 in the column of heights in the table, and against it you will find .111824; and .111824 × 30 X 30=100.6416.

2. The diameter of a circle is 100 feet, and the height of a segment is 6.5 feet; what is its area? Ans. 216.69 feet.

Solution. 1.065, and the tabular area for this height is .021669, which multiplied by the square of 100=216.69. The other exercises given above, may be performed in the same manner to exemplify this rule. When the area of a greater segment is required, find the tabular area of the less segment, and subtract it from .7854, and the remainder will be the tabular area of the greater segment.

1

A TABLE

OF THE AREAS OF THE SEGMENTS OF A CIRCLE,

The Diameter of which is Unity, and supposed to be divided into 100 equal parts.

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400293370

499 391699

401294350

.433 ⚫325900 ⚫466 ⚫358725
.434 ⚫326891 ⚫467 ⚫359723 ⚫500 ⚫392699

The areas of the circular zone, the lune, and the elliptic segments, may be readily found by means of this table; it will also be found particularly useful in calculating the solidities of the cylindric and conic ungulas, and the solidities of vaulted arches, &c.

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