This constant ratio is denoted by the Greek letter. But, by 795, this ratio for the circle with unit diameter, and therefore for every circle, is

802. For any circle.

FORMULA. C = 21π.

T = 3.141592+.

CIRCULAR MEASURE OF AN ANGLE.

803. When its vertex is at the center of the circle, by 506,

So, if we adopt as unit angle the radian, or that part of a perist. 4, that is, the angle subtended at the center

gon denoted by

T

A

of every circle by an arc equal to its radius, and hence named a radian, then

The number which expresses any angle in radians, also expresses its intercepted arc in terms of the radius.

If u denote the number of radians in any angle, and the length of its intercepted arc, then

The fraction arc divided by radius, or u, is called the circular measure of an angle.

804. Arcs are said to measure the angles at the center which include them, because these arcs contain their radius as often as the including angle contains the radian. In this sense an angle at the center is measured by the arc intercepted between its sides.

CHAPTER III.

MEASUREMENT OF SURFACES.

805. By 248, any parallelogram is equivalent to the rectangle of its base and altitude; therefore,

a

To find the area of any parallelogram.
RULE. Multiply the base by the altitude.
FORMULA. ᄆ o= ab.

806. COROLLARY. The area of a parallelogram divided by the base gives the altitude.

807. By 252, any triangle is equivalent to one-half the rectangle of its base and altitude; therefore,

Given, one side and the perpendicular upon it from the opposite vertex, to find the area of a triangle.

RULE. Take half the product of the base into the altitude.

808. Given, the three sides, to find the area of a triangle.

RULE. From half the sum of the three sides subtract each side separately; multiply together the half-sum and the three remainders. The square root of this product is the area.

FORMULA. A =

Vs (s

a) (s—b) (s—c). PROOF. Calling j the projection of c on b, by 306,

..

2hb = √(a + b + c) (b + c − a) (a + b − c) (a − b + c),

80g. To find the area of a regular polygon.

RULE. Take half the product of its perimeter by the radius of the inscribed circle.

PROOF.

Sects from the center to the vertices divide the polygon into congruent isosceles triangles whose altitude is the radius of the inscribed circle, and the sum of whose bases. is the perimeter of the polygon.

810. To find the area of a circle.

Ө

RULE. Multiply its squared radius by π.
FORMULA. O © = r2π.
по

If a regular polygon be circumscribed about the circle, its area N, by 809, is ≥rp.