MENSURATION.

Mensuration treats of the measurement of lines, surfaces, and volumes.

Important Suggestion.-Experience has shown that much, if not all, of the difficulty in mensuration results from the pupil's failure fully to understand the terms used in describing surfaces and solids, and from the consequent failure to get a clear conception of the objects themselves. Therefore it is suggested that pupils be required to learn all definitions. This can best be done by a careful study of the figures in connection with the definitions. Concrete illustration should be used whenever possible, and pupils should be permitted to handle objects. In the absence of geometrical forms, pupils should draw correct and neat figures to represent the conditions of each problem. Time thus spent will produce good results.

DEFINITIONS.

1. A Line has length, but no width.

2. A Straight Line is one which has the same direction throughout its whole length. It is the shortest distance between two points.

3. A Curved Line is one which changes its direction at every point in its length.

4. Parallel Lines are equidistant throughout their whole length.

5. A Horizontal Line is a line parallel to the horizon. The line A B is horizontal.

6. When two straight lines meet or intersect in such manner as to form right angles, they are said to be Perpendicular, the one to the other.

D

B

A

7. A Vertical Line is one that is perpendicular to the horizon. CO is a vertical line.

8. An Angle is the amount of divergence of two lines which meet at a point. The point is called the Vertex. In the angle A O C, O is the vertex.

The size of an angle is not dependent upon the length of the lines which form the angle.

9. There are three kinds of angles :

1. Right Angle.

2. Acute Angle, less than a right angle.

3. Obtuse Angle, greater than a right angle.

Draw an angle of each kind.

10. A Diagonal is a straight line joining opposite angles. 11. The Perimeter measures the bounding line of a surface.

12. An Inscribed Figure is the largest figure of a given kind that can be drawn within another. (See page 343.)

13. A Circumscribed Figure is the smallest figure of a given kind that can be drawn about another. (See page

343.)

14. Concentric Circles are those having the same centre. The space between two concentric circles is called a Ring. Draw two concentric circles.

SURFACES.

1. Surface is the outside of anything. Every surface has two dimensions,-length and breadth.

2. Area is the extent of a surface, and is estimated in square units; as, square inches, square feet, square yards, etc.

3. A Plane Surface is flat, like the walls and the floor of the school-room. Name some plane surfaces.

4. A Curved Surface is like that of a ball. Name some curved surfaces.

5. Surfaces are bounded by straight or curved lines; hence the terms rectilinear and curvilinear as applied to surfaces.

TRIANGLES.

1. A Triangle is a plane surface having three angles and three sides. Every triangle has two dimensions, altitude and base.

1

alt.

base

2. Triangles, classified according to their angles, are of three kinds :

1. Right Triangle, having one right angle.

2. Obtuse-Angled Triangle, having one obtuse angle. 3. Acute-Angled Triangle, having three acute angles. Draw a triangle of each kind.

3. Triangles classified according to their sides are of three kinds :

1. Equilateral Triangle,-all sides equal. 2. Isosceles Triangle,-two sides equal. 3. Scalene Triangle,-no two sides equal. Draw a triangle of each kind.

4. We have learned (page 164) that the area of a rectangle is the product of the length and breadth (base and altitude). Every triangle is regarded as one-half of a rectangle having the same base and altitude; hence the formula for the area of a triangle is :

base x altitude

By drawing figures and by cutting paper let pupils prove the foregoing.

PROBLEMS.

1. The base of a triangle is 150 yd. and its altitude is 75 yd. What is its area?

2. Required the area of a triangle whose base is 40 rd. and altitude 30 rd.

3. What is the area of an equilateral triangle whose sides are each 10 chains?

4. A board 5 ft. long has the shape of an isosceles triangle and measures at its base 15 inches. Find the number of square feet it contains.

5. Find the area of a right triangle, base 23.1 ft., altitude 32.1 ft.

alt.

base

Rhomboid.

PARALLELOGRAM.

1. A Parallelogram is a plane surface whose opposite sides are parallel.

2. There are four parallelograms:

1. Square-Sides parallel and equal; four right

angles.

2. Oblong-Sides parallel; opposite sides equal, adjacent sides unequal; four right angles.

3. Rhombus-Sides parallel and equal; two angles obtuse and two acute.

4. Rhomboid-Sides parallel; opposite sides equal; two angles obtuse and two acute.

3. The altitude of the Rhombus and the Rhomboid is the perpendicular distance between the parallel sides.

4. Make correct forms of the parallelograms. Draw the diagonal and mark the altitude.

5. The formula for the area of a parallelogram is:

1. A field in the form of a square is 64 rd. long. Find its area in acres.

2. How many square feet in an oblong board 90 in. long and 14 in. wide?

3. A pane has the form of a rhombus, measures 16 in. on each side, and the perpendicular distance between its sides is one-half the length of a side. Find its area.

4. Find the area in acres of a rhomboidal field which measures 10 ch. in length and 8 ch. in breadth.

TRAPEZOID.

1. A Trapezoid is a four

sided plane figure having two

sides parallel.

2. The altitude of a trape

alt.

zoid is the perpendicular distance between the parallel sides. 3. The formula for the area of a trapezoid is:

Area sum of parallel sides

2

X altitude.

What do you get when you divide the sum of the parallel sides by 2?

PROBLEMS.

1. Find the area of a trapezoid with parallel sides of 50 rd. and 78 rd., and with a distance between them of 391⁄2 rd.

2. A trapezoidal field contains 12 A. Its parallel sides are 220 rd. and 180 rd. How far apart are the parallel sides?