Incircle, also known as the inscribed circle, is a circle that is tangent to all sides of a polygon from the inside. It is the largest circle that can be inscribed within a polygon, touching all sides at exactly one point.
The concept of incircle involves several key knowledge points:
Polygons: In order to understand incircles, it is important to have knowledge about polygons. A polygon is a closed figure with straight sides, and it can be regular or irregular.
Tangency: Understanding the concept of tangency is crucial. A circle is said to be tangent to a line or a polygon when it touches it at exactly one point without intersecting it.
Radius: The radius of a circle is the distance from the center of the circle to any point on its circumference. In the case of an incircle, the radius is the distance from the center of the circle to any of the points where it touches the sides of the polygon.
Diameter: The diameter of a circle is the distance across the circle passing through its center. It is twice the length of the radius.
The formula to calculate the radius of the incircle (r) of a polygon with side length (s) is:
r = (A / s)
Where A is the area of the polygon.
To apply the incircle formula, follow these steps:
The symbol for incircle is a small circle drawn inside a polygon, touching all sides at exactly one point.
There are several methods to find the incircle of a polygon:
Using the formula: As mentioned earlier, the formula r = (A / s) can be used to calculate the radius of the incircle.
Using trigonometry: For certain polygons, such as triangles, trigonometric functions can be used to find the radius of the incircle.
Using the inradius: The inradius is the radius of the incircle. It can be found by dividing the area of the polygon by its semiperimeter (half the sum of all its sides).
Example 1: Find the radius of the incircle of a regular hexagon with a side length of 6 cm.
Solution: Step 1: Determine the area of the hexagon. The area of a regular hexagon can be calculated using the formula: A = (3√3 / 2) * s^2, where s is the side length. A = (3√3 / 2) * 6^2 = 54√3 cm^2
Step 2: Calculate the radius using the formula r = (A / s). r = (54√3 / 6) = 9√3 cm
Therefore, the radius of the incircle is 9√3 cm.
Example 2: Find the radius of the incircle of a triangle with side lengths 5 cm, 12 cm, and 13 cm.
Solution: Step 1: Determine the area of the triangle using Heron's formula. The semiperimeter (s) of the triangle is calculated as (a + b + c) / 2, where a, b, and c are the side lengths. s = (5 + 12 + 13) / 2 = 15 cm
Using Heron's formula, the area (A) of the triangle can be calculated as: A = √(s * (s - a) * (s - b) * (s - c)) A = √(15 * (15 - 5) * (15 - 12) * (15 - 13)) A = √(15 * 10 * 3 * 2) A = √900 = 30 cm^2
Step 2: Calculate the radius using the formula r = (A / s). r = (30 / 15) = 2 cm
Therefore, the radius of the incircle is 2 cm.
Question: What is the significance of the incircle in geometry? Answer: The incircle is significant in geometry as it helps determine various properties of polygons, such as their area, perimeter, and angles. It is also used in various geometric constructions and proofs. Additionally, the incircle plays a crucial role in the study of tangency and circle geometry.