geometry question about inscribed circles..help?

Question

a circle is inscribed in a rhombus whose diagals are 30cm and 40cm. what is the area between the rhombus and the circle?

could someone help pleasee? i dont understand how you can find the area with just the diagnols given ?

Answer 1

A rhombus has perpendicular diagonals, so the area formula A = (1/2)(d_1)(d_2) can be applied. Try to figure out why this works by cutting the rhombus into two triangles, along its diagonal. The area is 600 sq. cm.

As for the circle, its diameter is just the height of the rhombus, since it must touch opposite sides of the rhombus. First, find the length of the base. Since the diagonals of a rhombus also bisect each other, you can see that the base is the hypothenuse of a 15-20-25 right triangle. Then b = 25. Since the area is 600 and we know A = bh, h must be 24. Then the diameter is 24 and the radius is 12. The area of the circle is 144pi.

The area inside the rhombus but outside the circle is 600 - 144pi = 147.61.

Answer 2

(30/2)^2 + (40/2)^2 = c^2
15^2 + 20^2 = c^2
225 + 400 = c^2
c^2 = 625
c = 25

Ar = (pq)/2 = (30 * 40)/2 = 1200/2 = 600

x:20 = 15:25
x = 12

12 is the radius of the circle

A = pi * r^2
A = pi * 12^2
A = 144pi

600 - 144pi = about 147.6cm^2

ANS : about 148cm^2

For info, go to http://mathcentral.uregina.ca/QQ/database/QQ.09.00/jacky4.html

http://mathforum.org/dr.math/faq/formulas/faq.quad.html#rhombus

and

www.gomath.com/Questions/question.php?question=12828