Circumcenter Calculator

In this post, we will define circumcenter, discuss how to find circumcenter, and how to use our calculator to find circumcenter.

What is Circumcenter?

A circumcenter calculator is an online tool or application that calculates the circumcenter of a triangle based on its coordinates or side lengths. The circumcenter is the point where the perpendicular bisectors of the sides of a triangle intersect. It is also the center of the circle that passes through all three vertices of the triangle, which is called the circumcircle.

To use a circumcenter calculator, you typically need to input the coordinates of the three vertices of the triangle or the length of its sides. The calculator then uses mathematical formulas to calculate the coordinates of the circumcenter and, in some cases, other properties of the triangle such as its area, perimeter, and angles.

Circumcenter calculators are often used in geometry and trigonometry, as well as in computer-aided design and engineering applications where the properties of triangles and other geometric shapes are important. They can be found online as web-based tools or as part of larger software packages that include other mathematical and engineering functions.

How to find the circumcenter of a triangle?

1. Draw the triangle: Draw the triangle and label its vertices as A, B, and C.
2. Construct the perpendicular bisectors: For each side of the triangle, draw a line that is perpendicular to that side and passes through its midpoint. This line is called the perpendicular bisector of the side. You can use a straightedge and compass to do this.
3. Find the intersection point: The three perpendicular bisectors will intersect at a single point. This point is the circumcenter of the triangle.

Alternatively, you can use the following formula to find the coordinates of the circumcenter:

1. Calculate the midpoints: Find the midpoints of each side of the triangle. The midpoint of a side is the point that is halfway between its endpoints. If the coordinates of the vertices are A(x1, y1), B(x2, y2), and C(x3, y3), then the midpoints are:
   • M1 = ((x1 + x2)/2, (y1 + y2)/2)
   • M2 = ((x2 + x3)/2, (y2 + y3)/2)
   • M3 = ((x3 + x1)/2, (y3 + y1)/2)
2. Find the slopes: Find the slopes of the lines that pass through each side of the triangle. The slope of a line is the change in y divided by the change in x between any two points on the line. If the points (x1, y1) and (x2, y2) are on a line, then the slope is (y2 – y1)/(x2 – x1). The slopes of the lines that pass through sides AB, BC, and CA are:
   • m1 = (y2 – y1)/(x2 – x1)
   • m2 = (y3 – y2)/(x3 – x2)
   • m3 = (y1 – y3)/(x1 – x3)
3. Find the equations of the perpendicular bisectors: Find the equations of the lines that are perpendicular to each side of the triangle and pass through its midpoint. The equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. The equations of the perpendicular bisectors of sides AB, BC, and CA are:
   • y – (y1 + y2)/2 = (-1/m1) * (x – (x1 + x2)/2)
   • y – (y2 + y3)/2 = (-1/m2) * (x – (x2 + x3)/2)
   • y – (y3 + y1)/2 = (-1/m3) * (x – (x3 + x1)/2)
4. Find the intersection point: Solve the system of equations formed by the three perpendicular bisectors to find their point of intersection. This point is the circumcenter of the triangle.

Note that if the slopes of any two sides of the triangle are equal, then the triangle is isosceles and the perpendicular bisectors of those sides are parallel. In this case, the circumcenter can still be found using the other side.

Circumcenter calculator example

Suppose you want to find the circumcenter of a triangle with vertices A(1, 2), B(5, 6), and C(7, 2).

1. Go to a circumcenter calculator website or application. There are many available online, such as mathportal.org or calculatorsoup.com.
2. Input the coordinates of the three vertices of the triangle. In this case, you would enter A(1, 2), B(5, 6), and C(7, 2).
3. Click on the “Calculate” or “Find Circumcenter” button. The calculator will use mathematical formulas to find the circumcenter of the triangle.
4. View the results. The calculator may display the coordinates of the circumcenter, as well as other properties of the triangle such as its area, perimeter, and angles.

For example, using the mathportal.org circumcenter calculator, the circumcenter of the triangle with vertices A(1, 2), B(5, 6), and C(7, 2) is (4, 4), and the radius of the circumcircle is approximately 3.16.