A square is constructed with two of its sides on the binding radii and the remaining two vertices on the arc of the Sector of a disc of radius 12 units, the sectorial angle being 60°. Find the area of the square.

To tackle the problem of finding the area of the square inscribed in a sector of a circle, we need to visualize the setup and apply some geometric principles. The sector has a radius of 12 units and a central angle of 60 degrees. Let's break this down step by step.

Understanding the Geometry

First, we have a sector of a circle with a radius of 12 units and a central angle of 60 degrees. The two sides of the square are aligned along the radii of the sector, while the other two vertices touch the arc of the sector. This configuration allows us to use some trigonometric relationships to find the dimensions of the square.

Visualizing the Square

Imagine the sector as a slice of pizza. The two radii form the sides of the slice, and the arc is the crust. The square will sit inside this slice, with its base along the two radii. Let’s denote the side length of the square as \( s \).

Finding the Side Length

To find the side length \( s \), we can use the properties of the 30-60-90 triangle formed by the radii and the arc. The angle at the center of the circle is 60 degrees, which means that the angle between the radius and the line connecting the center to the vertex of the square is 30 degrees.

In a 30-60-90 triangle, the ratios of the sides are well-known: the side opposite the 30-degree angle is half the hypotenuse, and the side opposite the 60-degree angle is \( \sqrt{3}/2 \) times the hypotenuse. Here, the hypotenuse is the radius of the circle, which is 12 units.

• The height of the triangle (which corresponds to the side length \( s \) of the square) can be calculated as follows:

Using the sine function for the 30-degree angle:

\( s = 12 \cdot \sin(30^\circ) = 12 \cdot \frac{1}{2} = 6 \) units.

Calculating the Area of the Square

Now that we have the side length of the square, we can find its area. The area \( A \) of a square is given by the formula:

\( A = s^2 \).

Substituting the value of \( s \):

\( A = 6^2 = 36 \) square units.

Final Thoughts

Thus, the area of the square inscribed in the sector of the circle is 36 square units. This problem beautifully illustrates the interplay between geometry and trigonometry, allowing us to derive dimensions and areas using fundamental principles. If you have any further questions or need clarification on any steps, feel free to ask!