PLANIMETRY. 3 
surface). Ifa rectilineal figure is bounded by three lines, it is called a triangle ; 
if by four, a guadrilateral; if by more than four, a polygon. 
Triangles are divided, first, according to their sides, into equilateral (pl. 
1, fig. 15), in which all the sides are equal ; isosceles (fig. 16), which have 
only two sides equal; and scalene (fig. 17), in which all the sides are 
unequal. Secondly, they are divided with reference to their angles, into 
right angled triangles (fig. 18), when they have one angle right and two 
acute ; obtuse angled triangles (fig. 20), when they have one angle obtuse 
and two acute ; and acute angled triangles (fig. 19), when all the angles are 
acute. That side of a right angled triangle which is opposite to the right 
angle is called the hypothenuse, the two others are called the legs. 
Among quadrangular figures, the parallelograms form a remarkable class. 
They are quadrilaterals in which the two opposite sides are equal and parallel. 
Every parallelogram is divided by a diagonal—a straight line joining the 
vertices of two opposite angles—into two equal triangles (fig. 23). There 
are four different kinds of parallelograms: the square (fig. 21), in which all 
the sides are equal, and all the angles right angles; the rectangle (fig. 22), 
which has also its angles right angles, and only its parallel sides equal ; 
the rhombus or lozenge (fig. 24), in which all the sides are equal, but only 
its opposite angles equal; and the rhomboid (fig. 25), which has only its 
opposite sides and angles equal. A quadrilateral in which only two sides 
are parallel is called a trapezoid (fig. 27). It is called a right angled 
trapezoid when it has two right angles (jig. 26), and equilateral when the 
two sides that are not parallel are equal (fig. 28). A trapezoid may have 
three sides equal, but the parallel sides must always be unequal. A quadri- 
lateral in which none of the sides are parallel is called a trapezium 
( fig. 29). 
A polygon is called regular when all its sides and angles are equal, and 
irregular when this is not the case. Figs. 30-37 represent regular polygons, 
viz. fig. 30, a 5-sided figure, or pentagon; fig. 31, a 6-sided, or hexagon; 
jig. 32, a 7-sided figure, or heptagon; fig. 33, an 8-sided, or octagon ; fig. 34, 
a 9-sided figure, or nonagon ; fig. 35, a 10-sided, or decagon; fig. 36, one 
of 11 sides, or undecagon ; fig. 37, one of 12, or dodecagon ; all of which 
are accompanied by circles, either circumscribed or inscribed. 
The only curved line which occurs in elementary geometry, is the 
circular. The extremities of this line meet, and every point in it is equally 
distant from a point in the space inclosed, called the centre. The surface 
inclosed by the circular line is called the circle (fig. 38). In its relation to 
this, the circular line is called the circumference or periphery. A portion 
of the circular line is called an arc, e. g. abc (fig. 39). The size of an 
arc with reference to the whole circumference is measured by degrees. 
Kivery circle is divided into 360 equal parts, which are called degrees ; 
each degree contains 60 minutes, and each minute 60 seconds. A straight 
line, drawn from the centre of a circle to its circumference, is called a semt- 
diameter or radius, e.g. cd (fig. 40); a straight line uniting two points of 
the circumference, a chord, e. g. ef ( fig. 40, also fig. 43); and a diameter 
when, passing through the centre, it unites two opposite points of the 
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