All the vertices of a rhombus lie on a circle. If the area of the circle is \(1256\,\text{cm}^2\) (use \(\pi=3.14\)), find the area of the rhombus.
\(800\,\text{cm}^2\)
Step 1: Understand the property
If all vertices of a rhombus lie on a circle, then that rhombus must be a square (special property of cyclic quadrilaterals).
Step 2: Write the area formula of a circle
The area of a circle is given by:
\[ A = \pi R^2 \]
where \(R\) is the radius of the circle.
Step 3: Substitute the given values
\(1256 = 3.14 \times R^2\)
Step 4: Solve for \(R^2\)
Divide both sides by 3.14:
\[ R^2 = \dfrac{1256}{3.14} = 400 \]
Step 5: Find the radius
\[ R = \sqrt{400} = 20\,\text{cm} \]
Step 6: Relating radius to square
In a square inscribed in a circle, the diagonal of the square equals the diameter of the circle.
So, diagonal of square = \(2R = 40\,\text{cm}\).
Step 7: Find the side of the square
Diagonal of square = \( \sqrt{2} \times \text{side} \).
\[ \text{side} = \dfrac{40}{\sqrt{2}} = 20\sqrt{2}\,\text{cm} \]
Step 8: Find the area of the square
Area = (side)2
\[ = (20\sqrt{2})^2 = 400 \times 2 = 800\,\text{cm}^2 \]
Final Answer: The area of the rhombus is \(\boxed{800\,\text{cm}^2}\).