To find the area of a trapezium, multiply half the diagonal by the sum of the perpendiculars. This is an important and useful rule in all mensuration and land-surveying; when any rectilinear figure has to be measured (especially if it be an irregular one), the most convenient method for calculating B its area is to divide it into triangles, or into triangles and trapezoids; and having D found the area of each, to take their sum for the entire area. In this way, by means of a simple measuring line and cross-staff, it would be very easy to calculate the area of a field in the irregular shape of the figure ABCDEFG. E EXERCISES (F). Find the area of each of the trapezoids whose parallel sides, and the perpendicular distance between them, are respectively as follows: 1. 243 yds. and 131 yds.; perpendicular distance 352 yds. 2. 12 ft. and 7 ft.; 15.4 ft. 18 ft. 3 in. 53 ft. 6 in. 8. The diagonal of a trapezium is 42 yds. 3 in., and the perpendiculars are 19 yds. 1 ft. 6 in., and 21 yds. 2 ft. 3 in.; find the area. 9. Find the area of a trapezium whose sides, taken in order, are AB=192 yds.; BC=323 yds.; CD=156 yds.; DA=345 yds.; and the diagonal AC 438 yds. 10. The diagonal of a trapezium is 65 yds., and the two perpendiculars 28 yds. and 38.5 yds.; find the area. 11. The diagonal AC of a trapezium is 48 35 chains, and its sides are AB=4105 chains; BC+37 55 chains; CD=37.75 chains; DA= 27.4 chains; find the area. 12. A field in the form of a trapezium contains 16 ac. 3 ro. 8 po.; the diagonal is 16 chains, and the perpendiculars are in the ratio of 14 to 10; find the perpendiculars. LESSON X.-REGULAR POLYGONS. A regular polygon is a figure which has all its sides and all its angles equal. The name polygon (meaning many-cornered) would seem naturally to include none but figures with five sides or more, and to exclude the triangle and the square. Strictly speaking, however, they are all figures of the same class, and have common properties, and therefore the triangle and square are included in the following table. Regular polygons are called pentagons, hexagons, heptagons, octagons, etc., according as they have 5, 6, 7, or 8, etc., sides. A polygon may be regarded as divided from the central point into as many triangles as the polygon has sides. A line drawn from this point to the middle point of any one of the sides, will be the radius of the circle inscribed in the polygon, which is equal to the perpendicular height of each triangle. Now the area of each triangle is balf the base multiplied by the perpendicular height; and, therefore, the area of the whole polygon may be found by multiplying this by the number of sides. To find the area of a regular polygon, multiply half the perimeter by the radius of the inscribed circle. Without knowing the radius of the inscribed circle, the area of a polygon may be found from the length of one of its sides, to the square upon which it bears a constant ratio. In the following table, the third column gives the ratio which the radius of the inscribed circle bears to the side; and the fourth column gives the ratio of the area of the polygon to the square upon one side. In calculating the area by the former way, the radius of the inscribed circle must be found by multiplying the length of one side by the number in the third column; by the latter way, the area may be found at once by squaring the side, and multiplying it by the number in the fourth column. 1. Each side of a duodecagon measures 93 in.; find its area. 2. What is the cost of painting a heptagonal ceiling at 1s. 6d. per square yard, each side measuring 6 feet. 3. Find the area of a regular pentagon whose side measures 26 yds. 2 ft. 3 in. 4. The side of a regular pentagon measures 61 inches; find the radii of its inscribed and circumscribed circles. 5. The side of a pentagon measures 15 ft. 5 in.; find its area. 6. The side of a heptagon measures 35 in.; find its area. 7. Find the area of a field of octagonal shape, whose sides measure 12 chains each. 8. If the side of a pentagon is 5 yds., what is its area? Find the area of the following regular polygons : 9. Decagon with side 2050 links. We now pass on from rectilinear figures to those which are called curvilinear, being bounded not by straight lines, but by curved lines. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from its centre to the circumference are equal to one another. The straight line drawn from the centre of a circle to its circumference is called the radius. The straight line drawn from the circumference through the centre, until it meets the circumference again, is called the diameter, and is exactly equal to twice the radius. The diameter divides the circle into two semicircles. The rule for finding the area of a circle may be stated in various ways, all of which, however, are but different ways of stating one and the same thing. One of the simplest of them is as follows: To find the area of a circle, multiply half the circumference by the radius. You will no doubt notice that this is really the same rule as that for finding the area of a regular polygon. The circle may be regarded as a polygon with an innumerable number of sides. A triangle is very unlike a circle, but a square is somewhat nearer to it; a pentagon is nearer still; a hexagon and heptagon still nearer; a duodecagon is very much like a circle, and as the number of sides of the regular polygon increases, the figure becomes more and more circular, until at last the straight sides and the corners are no longer perceptible, and the figure is practically a circle. The rule for finding the area of a circle is only an adaptation of that for finding the area of a polygon, and may be stated in the same terms. The greater the number of sides which contain a polygon, the more nearly do the inscribed and circumscribed circles coincide with one another. This is exemplified by the foregoing table, which shows in order the ratio to the diameter of the circle of the perimeter of the circumscribed and inscribed polygons, beginning with a polygon of 4 sides, and proceeding by doubling the number of sides up to 8192. The above table is valuable, because it enables us to determine a very important ratio, viz., that of the circumference of a circle to its diameter. It is evident that the circumference of a circle is greater than that of the inscribed polygon, and less than that of the circumscribed. When the polygon does not contain many sides, the difference between these two circumferences is very perceptible; but as the number of sides is increased, it becomes less and less, until at last it is expressed only in the 7th decimal place, and is only one two-millionth part of the diameter; or, in other words, the exact circumference is less than 3.1415928 times, and more than 3.1415923 times the diameter. This difference is so infinitesimal as to be quite inappreciable, and we may therefore safely say that the circumference of a circle is 3.14159 times the diameter. Now, in the mensuration of all circular figures, this number constantly occurs, and is of the greatest importance, so that you must be careful to remember it, and not have to search for it in your book every time it is wanted. But as it would be troublesome to have to write down this number several times in a single calculation, mathematicians are accustomed, for convenience, to represent it by the Greek letter π. But as the diameter is exactly twice the radius, the circumference of a circle is-radius × 2 × 3.14159, or, as it is generally stated, circumference of circle = 2πr. The area and the circumference of a circle are dependent upon the length of its radius. Now the square (ABCD) of the diameter gives us more than the area of the circle, and must therefore be multiplied by a number somewhat less than unity to give it us. This number is really 7854, or T. |