CHAPTER X

MENSURATION

238. Mensuration (Latin mensura, a measure) is that branch of mathematics which treats of geometrical magnitudes. It is the application of arithmetic to geometry.

239. The four fundamental geometric concepts are the point, the line, the surface, and the solid. All but the point have magnitude.

1. A point is that which has position, but no magnitude.

2. A line is that which has only length. If a point moves, it generates a line.

3. A surface is that which has only length and width. If a line moves (not along itself), it generates a surface.

4. A solid is that which has length, width, and thickness. If a surface moves (not along itself), it generates a solid.

I. PLANE FIGURES

240. A plane is a surface such that any two points in it can be joined by a straight line which lies wholly in the surface.

241. A plane figure is any portion of a plane bounded by lines.

242. A polygon is a plane figure bounded by straight

lines. The perimeter of a polygon is the sum of the lines bounding it.

243. The least number of straight lines which can inclose a plane is three.

A polygon having three sides is a triangle; four sides, a quadrilateral; five sides, a pentagon; six sides a hexagon, etc.

244. A regular polygon is one whose sides and whose angles are equal.

245. The diagonal of a polygon is the straight line joining two angles not adjacent.

246. The base is the side upon which a figure is supposed to stand.

247. The altitude of a polygon is the perpendicular distance from the highest point to the line of the base.

248. The center of a regular polygon is the point within the polygon, equally distant from the middle points of the sides; and the apothem of such a polygon is the perpendicular line drawn from the center to the middle of a side.

249. A parallelogram is a quadrilateral whose opposite sides are parallel.

250. A rectangle is a parallelogram whose angles are right angles.

251. A square is a rectangle whose four sides are equal.

252. A trapezoid is a quadrilateral with only two sides parallel.

253. A trapezium is a quadrilateral with no two sides parallel.

254. A circle is a plane figure bounded by a curved line, every point of which is equally distant from the center. The boundary line of a circle is called the circumference. The diameter is a straight line drawn through the center of the circle, its

end points being in the circumference. The radius is a straight line drawn from the center to the circumference.

255. The area of a plane figure is its amount of surface. Area is denoted by the number of square units a figure contains.

256. The unit of surface is a square whose edge is a linear unit.

257. All surface measurements are based on the rectangle.

258. The fundamental principle is:

The area of a rectangle is (abstractly) equal to the product of the base by the altitude.

ILLUSTRATION

Find the area of a rectangle 5 in. long

and 3 in. wide.

3 in.

5 in.

The solution suggests the following

RULE. To find the area of a rectangle:

Multiply the base by the altitude.

NOTE.-For the sake of brevity, all rules of mensuration are ted as though the numbers were abstract.

259. The rule for finding the area of a parallelogram is e same as the rule for finding the area of a rectangle; cause the parallelogram can be converted into a recngle having the same base and the same altitude. Thus,

A

B

B

A

260. RULE. To find the area of a triangle: Take half the product of the base by the altitude. Reason: A triangle can be converted into a rectangle th the same base and half the altitude. Thus,

261. RULE. To find the area of a triangle when altitude is unknown, but the three sides are given:

Take the square root of the continued product of the sum of its sides and the remainders found by subtra each side from the half sum separately.

EXAMPLE

Find the area of a triangle whose sides are 6 in., 8 in., 10 in.

Let s
And a, b, c,

RELATION: Area

Area

.. Area

=

SOLUTION:

the sum of the sides, the sides, respectively.

√ s(s-a) (s—b) (s—c).

= V12 × 6 × 4 × 2 = 24.
24 sq. in.

262. RULE. To find the area of a trapezoid:

Take half the product of the sum of the bases by the altitud Reason: The trapezoid can be converted into a re tangle with a base equal to the sum of the bases of the trap zoid, and with an altitude equal to half the altitude of th trapezoid. Thus,

263. RULE.-To find the area of a trapezium and other

irregular polygons:

Divide the trapezium or other irregular polygon into triangles and find the sum of the areas of the triangles.