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WRITTEN EXERCISES.

1. What is the area of a trapezoid whose parallel sides are 160 feet and 100 feet respectively, and whose altitude is 80 feet?

2. What is the area of a trapezoid whose parallel sides are 140 and 80 feet, and whose height is 65 feet?

3. A field in the form of a trapezoid has two parallel sides, 75 and 100 rods respectively, and the distance between them is 60 rods: what is the area of the field? 4. Illustrate by an original problem the method of measuring a trapezoid.

ART. 486.-To find the area of a trapezium.—Divide the trapezium into two triangles by a diagonal, find the area of the triangles separately, and add the two together.

WRITTEN EXERCISES.

1. What is the area of a trapezium whose diagonal is 80 inches, and the altitudes of whose 2 triangles, measuring from the diagonal, are 40 and 50 inches respectively?

2. In a trapezium the sides in order are 30, 32, 34 and 36 feet, and the length of the shorter diagonal is 40 feet: what is the area? (Let the pupil draw the figure.)

3. Illustrate by an original problem the method of finding the area of a trapezium.

THE CIRCLE.

ART. 487.-A Circle is a plane figure bounded by a curved line called the circumference,' every point of which is equally distant from a point within called the center.

ART. 488.-The Diameter of a

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circle is a straight line beginning at the circumference,

passing through the center and ending in the circumference on the other side. Thus, AB is the diameter of the circle in the margin.

ART. 489.-The Radius of a circle is the distance from the center to the circumference. Thus, AC is the radius of the circle in the margin.

Illustration. Take as large a cylinder as can be obtained, measure its diameter and circumference accurately, and divide the circumference by the diameter. The quotient, if the measurements are accurate, is 3.1416, which is the ratio of the diameter to the circumference of any and every circle. Hence

ART. 490.-To find the circumference of a circle.-Multiply the diameter by 3.1416. To find the diameter of a circle.-Divide the circumference by 3.1416.

WRITTEN EXERCISES.

1. What is the circumference of a circle whose diameter is 30 inches?

2. If the circumference of a circle is 36 feet, what is the radius ?

3. Find the circumference of a circle whose diameter is 45 feet.

SUGGESTION.-Since the surface of a circle consists of a triangle whose base is the circumference and whose altitude is the radius, its area equals 3.1416 x Diameter of Diameter = D2 x .7854. (The convex surface of a cone is also a triangle. See Art. 505.) Hence

ART. 491. To find the area of a circle.-Multiply the square of the diameter by .7854.

WRITTEN EXERCISES.

1. What is the area of a circle whose diameter is 20 feet?

2. What is the area of a circle whose diameter is 30 inches?

3. What is the area of a circle whose radius is 10 feet?

4. What is the area of a circle whose circumference is 120 feet?

Measurement of Volumes.

ART. 492.-A Volume has length, breadth and thick

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ART. 495.-A Cylinder is a

Prism.

round volume of equal diameter Quadrangular
throughout, whose ends are cir-
cles.

ART. 496.-The Surface of a prism or of a cylinder is its entire surface less the surface of the two bases.

ART. 497.--The Altitude of a prism or of a cylinder is the perpendicular distance between the bases.

ART. 498.-To find the entire surface of

a prism or of a cylinder.-Multiply the perim

Pentagonal
Prism.

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eter of the base by the altitude: to find the entire surface, add the area of the bases.

WRITTEN EXERCISES.

1. What is the surface of a prism whose base is square, each of whose sides is 4 feet, and whose height is 15 feet?

2. What is the entire surface of the prism described in the foregoing question?

3. What is the convex surface of a cylinder 30 feet long and 10 feet in diameter ?

4. What is the entire surface of the cylinder described in the foregoing question?

5. Illustrate by an original problem the method of finding the surface of a prism.

6. Illustrate by an original problem the method of finding the surface of a cylinder.

ART. 499.-To find the contents of a prism or of a cylinder.-Multiply the area of the base by the altitude.

WRITTEN EXERCISES.

1. Find the contents of a square prism, each of whose sides is 5 feet and whose height is 7 feet.

2. Find the contents of a triangular prism, each of whose sides is 8 inches and whose height is 1 foot.

5. Find the contents of a cylinder 50 feet long and 6 feet in diameter.

4. Illustrate by an original problem the method of finding the contents of a prism or cylinder.

THE PYRAMID.

ART. 500.-A Pyramid is a volume bounded by a polygon and several triangles that meet at a common vertex.

ART. 501.- The polygon is the Base, and the triangles form the surface.

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ART. 502. To find the surface of a pyramid.- Multiply the perimeter of the base by one half the slant height.

WRITTEN EXERCISES.

1. What is the convex surface of a square pyramid whose base is 10 feet 6 inches, and whose slant height is 20 feet?

2. What is the convex surface of a triangular pyramid whose sides are each 5 feet, and whose slant height is 24 feet?

3. Illustrate by an original problem the method of finding the convex surface of a pyramid.

ART. 503.-To find the contents of a pyramid.-Multiply the area of the base by one third the altitude.

WRITTEN EXERCISES.

1. Find the contents of a pyramid whose base is 6 feet square, and whose altitude is 51 feet.

2. Find the contents of a pyramid whose base is a triangle, each side of which is 8 feet, and whose altitude is 40 feet.

3. Illustrate by an original problem the method of finding the contents of a pyramid.

THE CONE.

ART. 504. -A Cone is a volume

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whose base is a circle, and whose convex surface tapers uniformly to a point called the vertex.

ART. 505.-To find the convex surface of a cone.-Multiply the circumference of the base by one half the slant height.

WRITTEN EXERCISES.

1. Find the convex surface of a cone, the diameter of

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