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# Circumcenter Calculator

To find circumcenter of a triangle enter values in input box by using our Circumcenter Calculator.

Table of Contents:

## Circumcenter Calculator

Circumcenter calculator helps to calculate the circumcenter of a triangle.

## How circumcenter of a triangle calculator works?

The triangle circumcenter calculator** **calculates the circumcenter of triangle with steps**. **Follow these steps to find the circumcenter using the circumcenter finder.

- Enter the coordinates for points
and*A, B,* - Click the
**Calculate**button to see the result. - Use
**Reset**button to enter new values.

Circumcentre of a triangle calculator** **will instantly show you the circumcenter for the given coordinates.

## What is circumcenter?

Circumcenter of a triangle is the point of intersection of all three perpendicular bisectors of the triangle. Perpendicular bisectors are nothing but the line or a ray that cuts another line segment into two equal parts at 90 degrees.

The circumcenter's position depends on the type of triangle:

- the circumcenter lies inside the triangle (acute angle)
- the circumcenter lies outside the triangle (obtuse angle)
- the circumcenter lies at the center of the hypotenuse (right angle)

## How to find the circumcenter of a triangle?

Wondering how to calculate circumcenter** **without using circumcenter formula calculator?** **Well, there is no specific circumcenter formula** **to find it.

The circumcenter of a triangle can be found as the intersection of any two of the three perpendicular bisectors. The method to find circumcenter of triangle is given below.

- Find the midpoint of each side of the triangle.
- Find the slope of line.
- Calculate equation of line using slope and midpoint.

*y-y _{1} = m (x-x_{1})*

- Get the equation of all remaining lines using their midpoints and slope.
- Calculate intersection point by evaluating two bisector equations.
- The intersection point obtained in previous step is the circumcenter of triangle.

**Example:**

Find the circumcenter of triangle whose vertices are *(5, 4), (3, 1), (6, 1).*

**Solution:**

**Step 1: **First of all, we will calculate the midpoint for each line of triangle. We will have lines *AB, **and BC.*

Let,

*A = (5, 4)*

*B = (3, 1)*

*C = (6, 1)*

Suppose, ** D** = midpoint of

**and**

*AB***= midpoint of**

*E*

*BC.*__Midpoint of AB__

*D = **[(x _{1} + x_{2})/2, (y_{1} + y_{2})/2]*

*D = [(5 + 3)/2, (4 + 1)/2]*

*Midpoint of AB = D = (4, 5/2) *

__Midpoint of BC__

*E = [(x _{1} + x_{2})/2, (y_{1} + y_{2})/2]*

*E = [(3 + 6)/2, (1 + 1)/2]*

*Midpoint of BC = E = (9/2, 1)*

**Step 2: **Now, we will calculate the slope of line.

__Slope of AB__

Slope of *AB = [(y _{2} - y_{1})/(x_{2} - x_{1})] = [(1 - 4)/(3 - 5)]*

Slope of *AB = 3/2*

Slope of the perpendicular line to ** AB = -1/** Slope of

*AB = -1/(3/2) = -2/3*__Slope of BC__

Slope of *BC = [(y _{2} - y_{1})/(x_{2} - x_{1})] = [(1 - 1)/(6 - 3)]*

Slope of *BC = 0*

Slope of the perpendicular line to ** BC = -1/** Slope of

*BC = -1/0 = 0***Step 3: **Find the equation of line.

__Equation of the line AB__

*y = mx + b*

Where** m **refers to the slope of perpendicular line. Place the value of slope of perpendicular line

**in above equation.**

*AB, m = -2/3**y = -2/3(x) + b ------------>(1)*

Place the midpoint vertices of line *AB, *** D = (4, 5/2)** in equation 1.

*5/2 = -2/3(4) + b*

*b = 5.17*

Substitute ** b **in equation 1.

*y = -2/3(x) + 5.17 -------------->(2) *

__Equation of the line BC__

*y = mx + b*

Substitute the value of slope of perpendicular line ** BC, m = 0** in above equation.

*y = 0x + b *

*y = b ------------>(3)*

Place the midpoint vertices of line *BC, *** E = (9/2, 1)** in equation 3.

*b = 1*

Substitute ** b **in equation 3.

*y = 1 -------------->(4) *

**Step 4: **We have got the value of ** y. **Now we have to find the value of

**by substituting equation 4 in equation 2.**

*x**1 = -2/3(x) + 5.17 *

*x = -3/2(1 – 5.17)*

*x = -3/2(-4.17)*

*x = 6.25*

*So, x = 6.25 and y = 1*

*Circumcenter of triangle **∆**ABC = (6.25, 1)*

Don’t want to waste time doing all of the above stuff? Use the circumcenter solver** **above to save your precious time.

**References:**

- Circumcenter -- from merriam-webster
- Definition of Circumcenter. Mathsisfun.com.
- Triangle Circumcenter definition -Mathopenref.com.