Calculating Distance from the Center to a Side of an Inscribed Equilateral Triangle

An equilateral triangle with a side length of 20 cm is inscribed in a circle. This article will explore the steps to determine the distance from the center of the circle to one of the sides of the triangle. We will cover the geometric principles and calculations involved in finding the inradius, altitude, and circumradius.

Introduction to the Problem

Consider an equilateral triangle of side 20 cm inscribed in a circle. The task is to find the distance between the center of the circle and one of the sides of the triangle. This problem involves the relationship between the side length, inradius, and the circle's properties.

Step-by-Step Calculation

1. Inradius and Apothem

The inradius or apothem of an equilateral triangle is the distance from the triangle's center (incenter) to one of its sides. The formula for the inradius (r) in terms of the perimeter (P) is given by:

P 2 * 3√3 * r

Given that the perimeter (P) of the equilateral triangle is 60 cm (since (P 3 * 20)), the inradius (r) can be calculated as:

r P / (6√3) 60 / (6√3) 10 / √3 cm

2. Centre Angle and Side Length

The entire circle is divided into six equal sectors by the equilateral triangle. The center angle for each sector is 60 degrees. The half-side length of the triangle is 10 cm. Using the tangent function, we find the perpendicular distance from the center to one of the sides:

tan(60°) opposite side / adjacent side 10 / h

Thus, h 10 / tan(60°) 10 / √3 ≈ 5.77 cm

3. Circumradius

To find the circumradius (R), we start by considering the formula for the circumradius in an equilateral triangle:

R a^3 / (4A)

Where a is the side length and A is the area of the triangle. The area of an equilateral triangle is given by:

A (sqrt(3) / 4) * a^2

Substituting the side length of 20 cm:

A (sqrt(3) / 4) * (20)^2 (sqrt(3) * 400) / 4 100√3

Thus, the circumradius:

R (20^3) / (4 * 100√3) 8000 / (400√3) 20√3 / 3

Due to symmetry, the distance from the center to one of the sides is simply the inradius, which we have already calculated as 10 / √3 ≈ 5.77 cm.

4. Triangle Properties

We can also use the properties of the equilateral triangle and the right triangle formed by the height, half the base, and the side to find the altitude, which is also the distance from the center to the side:

In the right triangle ABM, where AB is the side of the triangle, BM is half the base, and AM is the altitude, we have:

AB^2 BM^2 AM^2

20^2 10^2 AM^2

AM^2 400 - 100 300

AM 10√3

The altitude AM from the center of the triangle to the side is one-third of this, as the center (circumcenter) divides the altitude in a 2:1 ratio:

OM AM / 3 10√3 / 3 ≈ 5.77 cm

Conclusion

The distance from the center of the circle to one of the sides of an inscribed equilateral triangle with a side length of 20 cm is approximately 5.77 cm. This value is consistent with the calculations performed using inradius, circumradius, and right triangle properties.