Inradius Calculator

Enter the three side lengths and area of your triangle to calculate the inradius, circumradius, incircle area, and triangle type.

Created by Luis GonzalezLast updated: May 9, 2026

How to Use This Calculator

1
Enter Side a
Input the length of the first side of your triangle, for example, 8 units.
2
Enter Side b
Provide the length of the second side of the triangle, such as 9 units.
3
Enter Side c
Input the length of the third side of the triangle, for instance, 10 units.
4
Enter the Area (A)
Input the known area of the triangle. If you don't have it, ensure the sides form a valid triangle and the area will be validated against Heron's formula.
5
Review your results
The calculator will display the inradius, semiperimeter, circumradius, incircle area, and even identify the type of triangle, along with a check for area consistency.

Example Calculation

A geometry student needs to find the inradius and related properties of a triangle with known side lengths and area.

Side a

Side b

Side c

Area (A)

34.197

Results

2.533111

Tips

Verify Triangle Validity

Before calculating, ensure the sum of any two sides is greater than the third side (e.g., a + b > c). If not, the sides cannot form a real triangle.

Accurate Area Input

If you provide the triangle's area, ensure it's calculated accurately. The tool cross-references it with Heron's formula based on side lengths, flagging inconsistencies.

Inradius and Triangle Shape

A larger inradius relative to the triangle's overall size generally indicates a more 'full' or equilateral-like shape, while a smaller inradius can suggest an elongated or obtuse triangle.

Calculating the Inradius and Geometric Properties of a Triangle

The Inradius Calculator allows you to determine the inradius, circumradius, incircle area, and other key geometric properties from a triangle's three side lengths and its area. For a triangle with sides 8, 9, and 10 units and an area of 34.197 square units, this tool computes the inradius as approximately 2.533 units, providing a comprehensive analysis of its inscribed and circumscribed circles. This is invaluable for students, engineers, and designers working with geometric shapes.

Why the Inradius Matters in Geometric Analysis

The inradius is a fundamental geometric property that provides deep insights into a triangle's shape and characteristics. It represents the radius of the largest circle that can be drawn entirely within the triangle, touching all three sides. This value is critical in various fields, from theoretical geometry proofs to practical applications in engineering design, such as optimizing the placement of circular components within triangular cavities. Understanding the inradius helps quantify a triangle's "compactness" and its relationship to other important centers and circles, influencing decisions in tessellations, packing problems, and even computer graphics.

The Relationship Between Area, Semiperimeter, and Inradius

The Inradius Calculator leverages the fundamental relationship between a triangle's area, its semiperimeter, and the inradius. The primary formula used is derived from the fact that the area of a triangle can be expressed as the sum of the areas of three smaller triangles formed by connecting the incenter to each vertex.

semiperimeter (s) = (side a + side b + side c) / 2
inradius (r) = Area (A) / semiperimeter (s)

circumradius (R) = (side a × side b × side c) / (4 × Area (A))
incircle area = π × r^2

Here, side a, side b, and side c are the lengths of the triangle's sides, and Area (A) is the total area. The semiperimeter (s) is half the perimeter, a common intermediate value in triangle calculations, and inradius (r) is the radius of the inscribed circle. The calculator also cross-references the input area with Heron's formula to ensure consistency.

💡 Understanding the angles within a triangle is crucial for many geometric calculations. Our Angle Between Two Vectors Calculator can help you determine these relationships in 2D or 3D space.

Example: Unpacking a Triangle's Inradius (Sides 8, 9, 10)

Let's analyze a triangle with Side a = 8, Side b = 9, Side c = 10, and an Area of 34.197 square units.

Calculate the Semiperimeter (s): s = (8 + 9 + 10) / 2 = 27 / 2 = 13.5 units.
Calculate the Inradius (r): r = Area / s = 34.197 / 13.5 = 2.533111 units.
Calculate the Circumradius (R): R = (8 × 9 × 10) / (4 × 34.197) = 720 / 136.788 = 5.2638 units.
Calculate the Incircle Area: Incircle Area = π × r^2 = π × (2.533111)^2 = 3.14159 × 6.4166 = 20.160 square units.
Determine Triangle Type: With sides 8, 9, 10, and checking c^2 vs a^2 + b^2 (100 vs 64+81=145), it's an "Acute triangle" (100 < 145).

The inradius is 2.533111 units, indicating a moderate inscribed circle.

💡 For more general mathematical operations, such as combining data sets, our Matrix Addition Calculator can assist with vector and matrix arithmetic.

The Geometry of Inscribed and Circumscribed Circles

In Euclidean geometry, the concepts of incircles and circumcircles are fundamental to understanding the intrinsic properties of triangles. The inradius (r) is a measure of the largest circle that can be inscribed within a triangle, tangent to all three sides, with its center (incenter) formed by the intersection of the angle bisectors. Conversely, the circumradius (R) is the radius of the circle that passes through all three vertices of the triangle, with its center (circumcenter) found at the intersection of the perpendicular bisectors of the sides. These radii are not merely theoretical constructs; they are crucial in fields like architectural design for optimal space utilization, in computer graphics for collision detection, and in advanced mathematical proofs, such as Euler's Theorem which elegantly relates the distance between the incenter and circumcenter.

Alternative Formulas for Calculating Inradius

While the formula r = A / s (Area divided by semiperimeter) is widely used and practical when the area is known or easily derivable, several alternative methods exist for calculating a triangle's inradius, each suited to different sets of known parameters.

One common variant, particularly useful for right triangles, simplifies to:

r = (a + b - c) / 2  // where c is the hypotenuse

This formula is derived from the properties of tangents from a vertex to the incircle.

Another general formula involves the angles of the triangle:

r = 4R × sin(A/2) × sin(B/2) × sin(C/2) // where R is the circumradius

This expression is more theoretical but demonstrates the deep connections between a triangle's radii and its angular properties. For practical application, the A/s formula is usually preferred unless angles or the circumradius are already known, or if the triangle is a right triangle.

Frequently Asked Questions

What is the inradius of a triangle?

The inradius (r) of a triangle is the radius of its incircle, which is the largest circle that can be inscribed inside the triangle, tangent to all three sides. The center of the incircle, known as the incenter, is the intersection point of the triangle's angle bisectors, making it equidistant from all three sides, a fundamental concept in Euclidean geometry.

How is the inradius different from the circumradius?

The inradius (r) is the radius of the incircle, which lies entirely within the triangle and is tangent to all three sides, while the circumradius (R) is the radius of the circumcircle, a circle that passes through all three vertices of the triangle. The incenter is found at the intersection of angle bisectors, whereas the circumcenter is at the intersection of perpendicular bisectors of the sides.

What is the significance of the inradius in geometry?

The inradius is significant in geometry as it relates a triangle's area (A) to its semiperimeter (s) through the formula A = r × s. It provides insights into the triangle's 'fullness' and is crucial in various geometric constructions, proofs, and optimization problems. For example, it helps determine the maximum size of a circular object that can fit within a triangular boundary.

Can the inradius be calculated without the triangle's area?

Yes, the inradius can be calculated without explicitly knowing the triangle's area if all three side lengths (a, b, c) are known. First, calculate the semiperimeter (s = (a + b + c) / 2), then use Heron's formula to find the area (A = sqrt(s(s-a)(s-b)(s-c))). Once the area is found, the inradius can be calculated using the formula r = A / s.