IF AREA OF A CIRCLE IS 144 MTRE SQ.,THEN WHAT IS THE AREA OF THE TRIANGLE FORMED INSIDE THE CIRCLE AT 45 DEGREE

To find the area of a triangle formed inside a circle when the area of the circle is given, we first need to determine the radius of the circle and then use that to find the area of the triangle. Let's break this down step by step.

Step 1: Calculate the Radius of the Circle

The area of a circle is calculated using the formula:

A = πr²

Where A is the area and r is the radius. Given that the area of the circle is 144 square meters, we can set up the equation:

144 = πr²

To find the radius, we rearrange the equation:

r² = 144/π

Now, taking the square root of both sides gives us:

r = √(144/π)

Calculating this, we find:

r ≈ 6.77 meters (using π ≈ 3.14 for simplicity).

Step 2: Understanding the Triangle Formed

When a triangle is formed inside the circle at a 45-degree angle, we can visualize it as an isosceles triangle where the two equal sides are the radius of the circle. The angle between these two sides is 45 degrees.

Step 3: Area of the Triangle

The area of a triangle can be calculated using the formula:

Area = 1/2 × base × height

In this case, we can also use the formula for the area of a triangle based on two sides and the included angle:

Area = 1/2 × a × b × sin(C)

Here, a and b are the lengths of the two sides (both equal to the radius), and C is the included angle (45 degrees).

Substituting the values:

Area = 1/2 × r × r × sin(45°)

Since sin(45°) = √2/2, we can rewrite the area as:

Area = 1/2 × r² × (√2/2)

Now substituting the value of r:

Area = 1/2 × (144/π) × (√2/2)

Calculating this gives:

Area ≈ 51.06 square meters (using π ≈ 3.14).

Final Thoughts

So, the area of the triangle formed inside the circle at a 45-degree angle is approximately 51.06 square meters. This process illustrates how we can connect different geometric properties and formulas to solve problems involving circles and triangles effectively.