1007. A Plane Figure is a plane surface bounded by lines either straight or curved.

1008. A Polygon is a figure bounded A Polygon of three sides is called a Triangle, of four sides, a Quadrilateral, etc.

by straight lines; as, ABCDE.

B

E

D C

1009. A Diagonal of a polygon is a line joining the vertices of two angles not consecutive. 1010. The Perimeter of a polygon is the sum of its sides.

1011. The Area of a plane figure is the number of square units in its surface.

NOTE. The principles of mensuration are derived from geometry; it is customary to give their application to practical purposes in arithmetic.

THE TRIANGLE.

1012. A Triangle is a polygon of three sides and three angles; as, ABC.

1013. The Base is the side upon which it seems to stand; as, AB. The Altitude is a line perpendicular to the base, drawn from the angle opposite; as, CD.

1014. A Right-Angled Triangle is a triangle which has one right angle; when one angle is obtuse, it is called obtuse-angled; when all the angles are acute, it is called acute-angled.

1015. An Equilateral Triangle is a triangle which has its three sides equal; when two sides are equal it is called isosceles; when its sides are unequal it is called scalene.

Rule. To find the area of a triangle, multiply the base by one-half of the altitude.

NOTE. If the three sides are given and not the altitude, take half the sum of the sides, subtract from it each side separately, multiply the half sum and these remainders together, and take the square root of the product.

1. The base of a triangle is 25 feet and the altitude 16 feet; what is the area? Ans. 200 sq. ft.

2. The base of a triangular lot is 370 yards and the altitude is 915 feet; how many acres does it contain?

Ans. 11.658+ acres.

3. It requires 252 square feet of boards to cover the gable of a house, and the height is 10 feet 6 inches; what is the base? Ans. 24 feet.

4. The three sides of a field are 56 rods, 72 rods, and 98 rods respectively; what is the area? Ans. 12 A. 70.28 P. 5. The perimeter of a piece of land in the form of an equilateral triangle is 624 rods; what is the area?

Ans. 177 A. 34.944 P.

THE QUADRILATERAL.

1016. A Quadrilateral is a polygon having four sides and therefore four angles. There are three classes, the parallelogram, trapezoid, and trapezium.

1017. A Parallelogram is a quadrilateral whose opposite sides are parallel. The altitude is the perpendicular distance between its opposite sides.

1018. A parallelogram which is rightangled is called a Rectangle. When the four sides are equal it is called a Square.

1019. An oblique-angled parallelogram is called a Rhomboid. An equilateral rhomboid is called a Rhombus.

Rule. To find the area of a parallelogram, multiply the base by the altitude.

1. What is the area of a parallelogram whose base is 35 feet and altitude 15 feet? Ans. 525 sq. ft.

2. What is the altitude of a rhomboid whose base is 63 inches and area 3087 sq. in?

Ans. 49 inches. 3. What is the difference in the area of two farms, one being 520 rd. long and 65 rd. wide, and the other 95 chains long and 45 chains wide?

Ans. 216 A. 40 P.

4. A carpenter had a plank 20 inches wide, from which he wished to saw off 10 square feet; what will be the length of the piece sawed off?

Ans. 6 ft.

1020. A Trapezoid is a quadrilateral

which has two of its sides parallel.

Its

allitude is the perpendicular distance be

tween its parallel sides.

Rule. To find the area of a trapezoid, multiply onehalf the sum of the parallel sides by the altitude.

1. Required the area of a trapezoidal garden, one side 52 ft., the other 75 ft., and altitude 40 ft.

2. What is the surface of a plank 18 25 in. at the other, and 16 ft. long?

Ans. 2540 sq. ft. in. wide at one end,

Ans. 283 sq. ft.

3. Owing to the irregularity of the streets, a house lot measured 42 ft. front, and only 33 ft. back; its depth was 50 ft.; what were the contents of the lot? Ans. 1875 sq. ft.

4. There are two fields, one a rectangle 56 feet long and 42 feet wide, and the other a trapezoid, one side being 80 feet and the other 88 feet, and the altitude 28 feet; what is the difference in their areas? Ans. They are equal.

1021. A Trapezium is a quadrilateral which has none of its sides parallel. A diagonal, AB, divides the trapezium into two triangles.

A

B

Rule. To find the area of a trapezium, divide the trapezium into two triangles by a diagonal, find the area of each triangle, and take the sum.

1. Required the area of a trapezium, whose diagonal measures 156 ft., and the altitudes of the two triangles are 45 and 54 feet respectively. Ans. 7722 sq. ft.

2. The diagonal of a tract of land in the form of a trapezium measures 75 chains, and the length of the sides are 35, 50, 70, and 85 chains respectively; what is the area? Ans. 322 A. 109.328 P.

REGULAR AND IRREGULAR POLYGONS.

1022. A Regular Polygon is one whose sides and angles are respectively equal.

Rule I. To find the area of a regular polygon, multiply

half the perimeter by the perpendicular let fall from the centre on one of the sides.

Rule II.-Square the side of the polygon, and multiply by the tabular area set opposite the polygon.

1. What is the area of a regular hexagon, whose side is 12 feet and perpendicular 10.39 feet? Ans. 374.04 sq. ft. 2. What is the area of a regular pentagon, whose side is 15 feet? Ans. 387.107+sq. ft.

3. A gentleman has an octagonal summer house, the side of which measures 9 feet; how much ground does it cover? Ans. 391.102 sq. ft.

1023. Rule. To find the area of an irregular polygon, draw diagonals dividing the polygon into triangles, find the area of these triangles, and take their sum.

1. What is the area of an irregular pentagon, whose diagonals are 125 and 130 inches, and the perpendiculars on the first diagonal are 20 inches and 35.7 inches, and on the second 20 inches? Ans. 4781.25 sq. in.

2. In a hexagonal field, the first side is 42 chains, the second 35, the third 27, the fourth 37, the fifth 35, the sixth 32; the diagonal from the first angle to the third is 49, from the first to the fourth 40, from the first to the fifth 45; what is the area? Ans. 251A. 134.368P.

THE CIRCLE.

1024. A Circle is a plane figure bounded by a curved line, every point of which is equally distant from a point within called the centre.

1025. The curved line is called the circumference, and a line passing through the centre and end

ing in the circumference is the diameter, as AB. diameter is called the radius, as BC or CD.

Half the

1026. An Arc is any part of the circumference, as AD or BD.

1027. A Chord is a straight line joining the extremities of an arc, as AD.

1028. A Segment is a portion of the circle included between an arc and its chord, as AED.

1029. A Sector is a portion of the circle included by an arc and the radii drawn to its extremities, as DCB.

Rule. To find the circumference of a circle, multiply the diameter by 3.1416.

1. What is the circumference of a flower-bed whose diameter is 36 inches ? Ans. 113.0976 inches. 2. If the diameter of the earth is 7912 miles, what is its circumference? Ans. 24856.339+ miles. 3. Which requires the most fence, a circular field 15 rods in diameter, or a square one whose side is 14 rods?

Ans. The latter, 8.876 rods. 1030. Rule.-To find the diameter of a circle, multiply

the circumference by .3183.

1. A carriage wheel is 7 feet in circumference; what is its diameter? Ans. 2.2281 feet.

2. A circular park is 3 miles in circumference; what is its diameter ? Ans. .9549 miles.

1031. Rule I.—The area of a circle equals the circumference multiplied by one-fourth of the diameter, or the square of the circumference multiplied by .07958.

Rule II.—The area of a circle equals the square of the radius multiplied by 3.1416, or the square of the diameter multiplied by .785398.

NOTE. The area will vary slightly, for the different rules.

1. The diameter of a circle is 16 and circumference 50.2656; what is the area? Ans. 201.0624.

2. The circumference of a circular pond is 144 feet; what is its area? Ans. 1650.17 sq. ft.