A square is constructed with two of its vertices on the bounding radii and wo remainng vertices on the arc of a sector of a disc of radius 12 units ,the sectorial angle being 60⁰, find the area of the square

To find the area of the square inscribed in a sector of a circle, we need to analyze the geometry of the situation. The sector has a radius of 12 units and a central angle of 60 degrees. We can visualize the square such that two of its vertices lie on the radii of the sector and the other two vertices touch the arc of the sector. Let’s break this down step by step.

Understanding the Geometry

First, we need to establish the coordinates of the key points in our sector. The sector can be visualized in the Cartesian plane, where the center of the circle is at the origin (0,0). The two radii of the sector will extend from the origin at angles of 0 degrees and 60 degrees.

Coordinates of the Radii

• The endpoint of the radius at 0 degrees is (12, 0).
• The endpoint of the radius at 60 degrees can be calculated using trigonometric functions:
  • x-coordinate: 12 * cos(60°) = 12 * 0.5 = 6
  • y-coordinate: 12 * sin(60°) = 12 * (√3/2) = 6√3
  So, the coordinates are (6, 6√3).

Positioning the Square

Let’s denote the side length of the square as 's'. The two vertices of the square on the radii will be at points (0, 0) and (s, 0) along the x-axis, while the other two vertices will be positioned at (s, s) and (0, s) along the y-axis. The vertices (0, s) and (s, s) must lie on the arc of the sector.

Finding the Relationship

For the vertex (s, s) to lie on the arc of the sector, it must satisfy the equation of the circle defined by the radius of the sector:

(x - 0)² + (y - 0)² = r²

Substituting the coordinates of the vertex (s, s) into the equation gives:

s² + s² = 12²

2s² = 144

s² = 72

s = √72 = 6√2

Calculating the Area of the Square

The area of a square is given by the formula:

Area = side length²

Substituting our value for 's':

Area = (6√2)² = 36 * 2 = 72 square units.

Final Thoughts

Thus, the area of the square inscribed in the sector of the circle with a radius of 12 units and a central angle of 60 degrees is 72 square units. This problem beautifully illustrates how geometry and algebra can work together to solve real-world problems involving shapes and their properties.