Area Between an Equilateral Triangle and Its Inscribed Circle: A Comprehensive Guide

Understanding the relationship between an equilateral triangle and its inscribed circle is a fundamental concept in geometry. This article explores how to calculate the area of the region between an equilateral triangle and its inscribed circle, providing a step-by-step solution and exploring the geometric principles involved. We will use an equilateral triangle with a side length of 8 cm as an example.

Step-by-Step Calculation

Step 1: Calculate the Area of the Equilateral Triangle

The area of an equilateral triangle with side length s is given by the formula:

A (frac{sqrt{3}}{4} s^2)

For a triangle with side length s 8 cm:

A (frac{sqrt{3}}{4} 8^2 frac{sqrt{3}}{4} times 64 16sqrt{3} , text{cm}^2)

Step 2: Calculate the Radius of the Inscribed Circle

The radius r of the inscribed circle of an equilateral triangle can be calculated using the formula:

r (frac{s sqrt{3}}{6})

For s 8 cm:

r (frac{8 sqrt{3}}{6} frac{4 sqrt{3}}{3} , text{cm})

Step 3: Calculate the Area of the Inscribed Circle

The area A_c of the circle is given by the formula:

A_c (pi r^2)

Substituting the value of r:

A_c (pi left(frac{4 sqrt{3}}{3}right)^2 pi left(frac{16 cdot 3}{9}right) frac{48pi}{9} frac{16pi}{3} , text{cm}^2)

Step 4: Calculate the Area Between the Triangle and the Circle

The area between the triangle and the circle A_{area} is given by:

A_{area} A - A_c

Substituting the areas we found:

A_{area} (16sqrt{3} - frac{16pi}{3} , text{cm}^2)

Alternative Method Using Geometry and Trigonometry

To find the area of the region between the inscribed circle with center O and the equilateral triangle ABC with each side 8 cm, we can use a different approach.

Step 1: Calculate the Area of Right Triangle OPB

Firstly, we can calculate the area of the right triangle OPB. In a 30-60-90 triangle, the ratio of the sides is 1: √3: 2. Given OB 4 cm, we have:

(tan 30^circ frac{OP}{4} frac{1}{sqrt{3}})

OP frac{4}{sqrt{3}} , text{cm})

The area of ΔOPB is given by:

Area of ΔOPB (frac{1}{2} times 4 times frac{4}{sqrt{3}} frac{8}{sqrt{3}} , text{cm}^2)

Step 2: Calculate the Area of Sector OSP

The area of sector OSP is given by:

Area of sector OSP (frac{60}{360} pi r^2 frac{1}{6} pi left(frac{4 sqrt{3}}{3}right)^2 frac{1}{6} pi left(frac{16 cdot 3}{9}right) frac{16pi}{18} frac{8pi}{9} , text{cm}^2)

Step 3: Calculate the Area Between the Triangle and the Circle

The area between the triangle and the circle is the difference between the area of the sector and the area of the triangle OPB:

Area between triangle and circle (frac{8pi}{9} - frac{8}{sqrt{3}} , text{cm}^2)

Conclusion

By understanding the geometric relationships and applying the appropriate formulas, we can accurately calculate the area between an equilateral triangle and its inscribed circle. This method not only provides a precise result but also offers insight into the interplay between different shapes and their properties.