The degree measure of the rhombus angle is 60 °. Calculate the area of a circle

The degree measure of the rhombus angle is 60 °. Calculate the area of a circle inscribed in this rhombus if the length of its smaller diagonal is 6 cm.

1. It is known:

a) the diagonals of the rhombus divide the corners of the rhombus in half and at the point of intersection are perpendicular and halved;

b) the area of ​​the circle is equal to P * R ^ 2;

c) the radius of the circle inscribed in the rhombus R = d1 * d2: 4 * a, where d1 and d2 are the diagonals of the rhombus, and is its side.

2. In a right-angled triangle formed by the halves of the diagonals and the side of the rhombus, we find half of the second diagonal d2, if it is known that d1 = 6 cm.

d1 / 2: d2 / 2 = tg30 *;

d2 / 2 = d1 / 2: tg30 * = 3: 3 ^ 1/2: 3 = 9: 3 ^ 1/2;

d2 = 18: 3 ^ 1/2.

3. By the Pythagorean theorem, from the same triangle we calculate the side of the rhombus a, which in the triangle is the hypotenuse.

a ^ 2 = (d1 / 2) ^ 2 + (d2) ^ 2 = 9 + (9: 3 ^ 1/2) ^ 2 = 9 + 27 = 36;

a = 36 ^ 1/2 = 6 cm.

4. Calculate the radius R of the inscribed circle.

R = d1 * d2: 4 * a = (6 * 18: 3 ^ 1/2): 4 * 6 = 4.5: 3 ^ 1/2 cm.

5. Find the area S of the inscribed circle.

S = P * R ^ 2 = 3.14 * 4.5 ^ 2: 3 = 3.14 * 20.25: 3 = 21.2 cm ^ 2.

Answer: The area of ​​the inscribed circle is 21.2 square centimeters.