If the ratio of a circle's sector to its total area is 78 what is the measure of its sector's arc?

Length of arc = angle of arc (in radians) × radius of circle

With a ratio of 7:8 the area of the sector is 7/8 the area of the whole circle.

This is the same as saying that the circle has been divided up into 8 equal sectors and 7 have been shaded in.

Dividing the circle up into 8 equal sectors will give each sector an angle of arc of 2π × 1/8

7 of these sectors will thus encompass an angle of arc of 2π × 1/8 × 7 = 2π × 7/8 = 7π/4

Thus the length of the arc of the sector is 7π/4 × radius of the circle.

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Alternatively, it can be considered that as 7/8 of the area is in the sector, the length of the arc is 7/8 the circumference of the circle = 7/8 × 2π × radius = 7π/4 × radius.