Introduction

This article provides a detailed step-by-step guide on how to calculate the area inside a circle that includes the area of an inscribed equilateral triangle. Understanding these geometric principles is essential for students and professionals working with spatial mathematics. We'll break down the problem into manageable parts and explore each step in detail.

Step-by-Step Calculation

Step 1: Calculating the Area of the Equilateral Triangle

Given an equilateral triangle with a side length of 33 cm, we can use the formula for the area of an equilateral triangle to find the area of the triangle.

Formula

The area, ( A ), of an equilateral triangle with side length ( s ) is given by:

[ A frac{sqrt{3}}{4} s^2 ]

Substituting ( s 33 ) cm into the formula, we get:

[ A frac{sqrt{3}}{4} times 33^2 frac{sqrt{3}}{4} times 1089 frac{1089 sqrt{3}}{4} approx 471.7 , text{cm}^2 ]

Step 2: Calculating the Radius of the Circumscribed Circle

The radius, ( R ), of the circumscribed circle around an equilateral triangle can be calculated using the formula:

[ R frac{s}{sqrt{3}} ]

Substituting ( s 33 ) cm into the formula, we get:

[ R frac{33}{sqrt{3}} 11 sqrt{3} , text{cm} ]

Step 3: Calculating the Area of the Circle

The area, ( A_c ), of the circle is given by the formula:

[ A_c pi R^2 ]

Substituting for ( R ), we get:

[ A_c pi times (11 sqrt{3})^2 pi times 121 times 3 363 pi , text{cm}^2 ]

Calculating the area of the circle:

[ A_c approx 363 times 3.14159 approx 1147.2 , text{cm}^2 ]

Step 4: Total Area Inside the Circle Including the Triangle

The total area inside the circle including the area of the triangle is calculated by subtracting the area of the triangle from the area of the circle and then adding the area of the triangle back in (since it was subtracted once):

[ text{Total Area} A_c - A A A_c approx 1147.2 , text{cm}^2 ]

Therefore, the total area inside the circle including the area of the triangle is approximately:

[ boxed{1147.2 , text{cm}^2} ]

Visual and Practical Insights

The figure below shows the process of constructing the equilateral triangle and the circumscribed circle. Always draw a picture first when faced with such problems.

Construction Steps

1. Draw a line of length 33 cm and set your compass to this distance/spread.

2. This line forms one of the sides of the triangle. Create two arcs of 33 cm centered at the ends of this line segment to find the other two vertices.

3. Use the upper intersection point to be a vertex of the triangle. Draw the other two sides to the ends of the line segment.

4. To construct the circumscribing circle, find the intersection point of the perpendicular bisector of one side and the perpendicular bisector of another side.

5. Set the compass to the distance from this intersection point to any of the triangle vertices.

Conclusion

Mastering these geometric principles will significantly enhance your problem-solving skills in mathematics and related fields. By following these steps and practicing similar problems, you can confidently handle complex spatial and geometric calculations.