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Elementary Geometry for College St...
Solutions
Elementary Geometry for College St...
Suppose a circle of radius r is inscribed in a rhombus, each of whose sides has length s. Find an expression for the area of the rhombus in terms of r and s.
To find:
An expression for area of rhombus
Formula:
Area of triangle:
If b and h are lengths of base and height of triangle respectively, then area A of triangle is given by the formula,
Properties of rhombus:
Rhombus is a parallelogram with all equal sides.
The diagonals of a rhombus are perpendicular bisectors of each other.
Calculation:
Consider the following diagram in which ABCD is a rhombus with an inscribed circle.
It is given that length of side of rhombus is s. We know that all sides of a rhombus are equal. Hence, in the above diagram,
Let’s find the area of
For finding the area, let’s consider AD as the base of triangle. Thus, base
If AD is the base, OE is the corresponding height. It is clear from the diagram that OE is the radius of circle. The radius of circle is given as r. Thus, height of
Area of
Area of
Consider
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