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Show that the area of a zone of one base is equal to the area of the circle whose radius is the chord c of the generating arc AB of the zone.
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| Photo by Math Principles in Everyday Life |
Solution:
A spherical zone or zone, in short is a portion of the surface of a sphere from its circular cross section to its end (for one base) or between two parallel circular planes (for two bases). The above figure is a zone of one base.
In this problem, we will compare the area of a zone with the area of a circle whose radius is chord AB or c. Let's see if they will be equal or not. Analyze and label further the given figure above as follows
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| Photo by Math Principles in Everyday Life |
Apply Pythagorean Theorem for ∆OCB,
Apply Pythagorean Theorem for ∆ABC,
The area of a Zone is equal to
The area of a circle whose radius is chord AB or c is equal to
Therefore,
Area of Zone = Area of Circle
