Inscribed Circle Radius: Explained

by Wholesomestory Johnson 35 views

Hello there! This is a great question. You're asking about finding the radius of a circle that perfectly fits inside a triangle, also known as an incircle. I'll provide you with a clear, detailed, and correct answer. Let's dive in!

Correct Answer

The radius (r) of the circle inscribed in a triangle can be calculated using the formula: r = A/s, where A is the area of the triangle and s is the semi-perimeter (half the perimeter) of the triangle.

Detailed Explanation

Finding the radius of an inscribed circle is a common problem in geometry. It helps us understand the relationships between a triangle's sides, area, and the circle that fits snugly inside. Here's a breakdown:

Key Concepts

Let's define some key terms:

  • Incircle: The circle that lies inside a triangle and touches all three sides. The center of this circle is called the incenter.
  • Radius (r): The distance from the incenter to any side of the triangle. This distance is always perpendicular to the side.
  • Area (A): The space enclosed by the triangle. The area can be calculated using different formulas depending on the information we have (e.g., base and height, or using Heron's formula).
  • Semi-perimeter (s): Half of the perimeter of the triangle. If the sides of the triangle are a, b, and c, then s = (a + b + c) / 2.

Formula Breakdown

The core formula to remember is: r = A/s

Here's how it works:

  1. Finding the Area (A):

    • The area of a triangle can be found using various formulas. The choice of formula depends on what information you have. Common formulas include:
      • If you know the base (b) and height (h): A = (1/2) * b * h
      • If you know all three sides (a, b, c): Use Heron's formula: A = sqrt(s * (s - a) * (s - b) * (s - c)), where s is the semi-perimeter.

  2. Finding the Semi-perimeter (s):

    • Calculate the perimeter by adding the lengths of all three sides: Perimeter = a + b + c.
    • Divide the perimeter by 2 to get the semi-perimeter: s = Perimeter / 2 or s = (a + b + c) / 2.

  3. Calculating the Radius (r):

    • Once you have the area (A) and the semi-perimeter (s), plug these values into the formula: r = A/s.

Step-by-Step Example

Let's work through an example to solidify the concept:

Problem: A triangle has sides of length 5 cm, 12 cm, and 13 cm. Find the radius of the inscribed circle.

Solution:

  1. Find the Semi-perimeter (s):

    • s = (a + b + c) / 2
    • s = (5 + 12 + 13) / 2
    • s = 30 / 2
    • s = 15 cm

  2. Find the Area (A):

    • Since this is a right-angled triangle (because 5² + 12² = 13²), we can use the formula A = (1/2) * b * h.
    • A = (1/2) * 5 * 12
    • A = 30 cm²

  3. Calculate the Radius (r):

    • r = A/s
    • r = 30 / 15
    • r = 2 cm

Therefore, the radius of the inscribed circle is 2 cm.

Using Heron's Formula for Area

Let's look at another example where you don't immediately know the height and need to use Heron's formula:

Problem: A triangle has sides of length 7 cm, 8 cm, and 9 cm. Find the radius of the inscribed circle.

Solution:

  1. Find the Semi-perimeter (s):

    • s = (a + b + c) / 2
    • s = (7 + 8 + 9) / 2
    • s = 24 / 2
    • s = 12 cm

  2. Find the Area (A) using Heron's Formula:

    • A = sqrt(s * (s - a) * (s - b) * (s - c))
    • A = sqrt(12 * (12 - 7) * (12 - 8) * (12 - 9))
    • A = sqrt(12 * 5 * 4 * 3)
    • A = sqrt(720)
    • A ≈ 26.83 cm² (approximately)

  3. Calculate the Radius (r):

    • r = A/s
    • r = 26.83 / 12
    • r ≈ 2.24 cm (approximately)

Therefore, the radius of the inscribed circle is approximately 2.24 cm.

Special Cases and Considerations

  • Equilateral Triangles: In an equilateral triangle (where all sides are equal), the incenter, circumcenter, centroid, and orthocenter all coincide at the same point. The radius of the inscribed circle in an equilateral triangle with side 'a' is given by r = a / (2 * sqrt(3)).
  • Right Triangles: As we saw in the example, right triangles have a special relationship. The inradius can also be found using the formula: r = (a + b - c) / 2, where a and b are the lengths of the legs, and c is the length of the hypotenuse.
  • Obtuse Triangles: The same formulas apply to obtuse triangles (where one angle is greater than 90 degrees). The only difference is how you calculate the area, which might require using trigonometry.

Why is the formula r = A/s correct?

The formula r = A/s is derived from the following:

  1. Dividing the Triangle into Smaller Triangles: Imagine drawing lines from the incenter to each vertex of the triangle. This divides the triangle into three smaller triangles. The area of the original triangle is the sum of the areas of these three smaller triangles.
  2. Area of Each Smaller Triangle: Each smaller triangle has the inradius (r) as its height, and one side of the original triangle as its base.
  3. Total Area:
    • Area of first small triangle: (1/2) * a * r
    • Area of second small triangle: (1/2) * b * r
    • Area of third small triangle: (1/2) * c * r
    • Total Area, A = (1/2) * a * r + (1/2) * b * r + (1/2) * c * r
    • A = r * (a + b + c) / 2
    • Since (a + b + c) / 2 is the semi-perimeter (s), we get A = r * s.
    • Rearranging, we get r = A/s.

Key Takeaways

  • The radius of an inscribed circle is found using the formula: r = A/s.
  • 'A' is the area of the triangle.
  • 's' is the semi-perimeter (half the perimeter).
  • Find the area using the appropriate formula based on the information provided (base/height, sides).
  • Calculate the semi-perimeter by adding the sides and dividing by 2.
  • Remember to use the correct units (e.g., cm, m, inches) for your answer.
  • The incenter is the center of the inscribed circle, and it is equidistant from all three sides.

I hope this comprehensive explanation helps you understand how to find the radius of an inscribed circle! If you have any more questions, feel free to ask.