Relation between circumradius and inradius in different triangle

Relation between circumradius and inradius in different triangle

Aditya Raj Anand

Wednesday, 20 January 2021

math

Circumradius and inradius these two terms come from geometry. there is also a unique relation between circumradius and inradius. But relation depends on the condition or types of the polygon. like, if the polygon is square the relation is different than the triangle.

So, we can say that relation between circumradius and inradius will be different for different polygon. for example relation between circumradius and inradius of triangle is entirely different to the relation for equilateral triangle.

Before we discuss circumradius and inradius relation for different polygon, we should know the meaning of these two terms circumradius and inradius in detail. So lets start with Circumradius.

Circumradius is the radius of those circle which is circumscribe (surrounds) the triangle. for example,

consider a triangle ABC. Draw the bisector of all three line segment AB, BC, and AC respectively.

Remember all three bisector passes through the vertices A, B, and C of triangle ABC. Now, take as a center 'O' where all three bisector cuts each other. With center 'O' draw a circle touching all three vertices A, B and C. As shown in figure.

Now, OR = OQ = OB = OP = OA = OC = R (circumradius)

R is known as circumradius. 'O' is known as circumcenter of the circle.

Formula of circumradius is R = abc ∕ 4Δ

Inradius is the radius of those circle which is inscribe (surrounded) by the triangle. For example,

Consider a triangle ABC. Draw the bisector of angle A, B and C respectively. Take point 'O' as center where all three bisectors meet each other. With center 'O' draw a circle touching all three sides of the triangle ABC. As shown in figure.

As we know that,

Inradius is that radius of the circle which is inscribe (surrounded) by the triangle. In other words , those line segment which is perpendicular to the sides of the triangle with center 'O'. Here OR, OQ, and OP are perpendicular to the sides AB, AC, and BC respectively. So, OR = OP = OQ = Inradius.

Now, 

OR = OP = OQ = r

'r' is known as inradius of the circle.

'O' is known as incenter of the circle.

Formula of Inradius is, r = (a + b - c) / 2

Understanding the Relation Between Circumradius and Inradius in Different Triangles

In geometry, triangles play an essential role, and their properties have been studied in depth for centuries. One of the interesting aspects of a triangle is the relationship between its circumradius (denoted as $R$) and its inradius (denoted as $r$). These two radii represent the radii of the circumcircle and the incircle, respectively. They are important in understanding the triangle's overall structure and its relationship to various geometric properties.

In this article, below, we will discuss into the relation between circumradius and inradius in different types of triangles, including the right-angle triangle, equilateral triangle, isosceles triangle, and scalene triangle. Our goal is to break down these concepts into simple, understandable terms, making the relationship clear for both beginners and advanced learners.

Whether you are a student learning about triangle properties or a researcher exploring advanced geometric concepts, this relationship is a vital part of understanding triangle geometry. By mastering these concepts, you can gain a deeper appreciation for the elegance and interconnectedness of geometric shapes.

1. Relation Between Circumradius and Inradius in a Right-Angle Triangle

In a right-angled triangle, the relationship between the circumradius and the inradius is straightforward and easy to understand.

• Circumradius in a Right-Angle Triangle: In a right-angled triangle, the circumcenter (the center of the circumcircle) lies at the midpoint of the hypotenuse. Therefore, the circumradius ($R$) is half the length of the hypotenuse. If the sides of the triangle are $a$, $b$, and $c$, with $c$ being the hypotenuse, then:

  $$R = \frac{c}{2}$$

• Inradius in a Right-Angle Triangle: The formula for the inradius ($r$) in a right-angled triangle is given by:

  $$r = \frac{a + b - c}{2}$$

  Where $a$, $b$, and $c$ are the lengths of the sides of the triangle.

Thus, the circumradius and inradius in a right-angled triangle are related, but their values depend on the lengths of the sides of the triangle, especially the hypotenuse.

Example to demonstrate the relation between circumradius and inradius in right angle triangle.

Consider a right-angled triangle with sides $a = 3$, $b = 4$, and $c = 5$ (a Pythagorean triple).

• The circumradius is:

  $$R = \frac{c}{2} = \frac{5}{2} = 2.5$$

• The inradius is:

  $$r = \frac{a + b - c}{2} = \frac{3 + 4 - 5}{2} = 1$$

In this case, the circumradius is 2.5, and the inradius is 1.

2. Relation Between Circumradius and Inradius in an Equilateral Triangle

Equilateral triangles have special properties that make the relationship between the circumradius and inradius particularly elegant. In an equilateral triangle:

• Circumradius: The circumradius $R$ of an equilateral triangle is related to the side length $a$ by the formula:

  $$R = \frac{a}{\sqrt{3}}$$

• Inradius: The inradius $r$ of an equilateral triangle is given by:

  $$r = \frac{a}{2\sqrt{3}}$$

Notice that in an equilateral triangle, the circumradius is exactly $\sqrt{3}$ times larger than the inradius.

Example to demonstrate the relation between circumradius and inradius in an equilateral triangle.

For an equilateral triangle with side length $a = 6$:

• The circumradius is:

  $$R = \frac{6}{\sqrt{3}} = 2\sqrt{3} \approx 3.46$$

• The inradius is:

  $$r = \frac{6}{2\sqrt{3}} = \sqrt{3} \approx 1.73$$

Thus, in an equilateral triangle, the circumradius is $\sqrt{3}$ times the inradius.

Relation between circumradius and inradius of equilateral triangle in short

Relation between circumradius and inradius of an equilateral triangle is in such a way that Inradius of a circle is equal to the half of the Circumradius of a circle. As shown in above figure. the formula of inradius for equilateral triangle is same as formula of circumradius. But there is difference of 2 in denominator of inradius's formula. that is given by,

formula of inradius of equilateral triangle = side / 2√3

formula of circumradius of equilateral triangle is side / √3

Inradius = 1/2 ✕ circumradius

∴ circumradius = 2 inradius

3. Relation Between Circumradius and Inradius in an Isosceles Triangle

In an isosceles triangle, the circumradius and inradius can be derived using formulas that take into account the equal sides of the triangle. While the exact relationship may vary based on the specific dimensions of the isosceles triangle, there is a general pattern that can be followed.

• Circumradius: In an isosceles triangle, the circumradius can be found using the formula:

  $$R = \frac{a}{2\sin(A)}$$

  Where $a$ is the length of the equal sides, and $A$ is the angle between them.

• Inradius: The inradius for an isosceles triangle can be calculated as:

  $$r = \frac{a \sin(A)}{2}$$

Thus, in an isosceles triangle, the circumradius and inradius are both dependent on the side length $a$ and the angle $A$. These relationships help in understanding the geometric properties of the triangle.

4. Relation Between Circumradius and Inradius in a Scalene Triangle

In a scalene triangle, the circumradius and inradius do not have as simple a relationship as in right-angled or equilateral triangles. However, formulas still exist to calculate both radii.

• Circumradius: The formula for the circumradius in a scalene triangle is the same as in any triangle:

  $$R = \frac{abc}{4K}$$

  Where $a$, $b$, and $c$ are the side lengths of the triangle, and $K$ is the area of the triangle.

• Inradius: Similarly, the formula for the inradius is:

  $$r = \frac{A}{s}$$

  Where $A$ is the area, and $s$ is the semi-perimeter.

Although these formulas are more complex than those in the other types of triangles, they are still essential for calculating the circumradius and inradius in any scalene triangle.

How the Circumradius and Inradius are Connected

While the specific formulas for circumradius and inradius vary between different types of triangles, one key point is that both radii depend on the sides of the triangle, and they are both important in understanding the properties of the triangle as a whole.

For example, in many cases, the circumradius is related to the triangle's area and the lengths of its sides. The inradius, on the other hand, is connected to the area and semi-perimeter. In special types of triangles, such as right-angled or equilateral triangles, the formulas simplify, and the relationship between these two radii becomes clearer and easier to compute.