Circle Inscribed in a Triangle – Math Lessons


Greetings and welcome to MathSux! This post explores how to construct a circle inscribed in a triangle using a compass and straightedge. First we will find the Angle Bisector of each vertex of triangle ABC, then we will locate the incenter, measure the radius and use our compass to sketch and inscribe a circle into our triangle. If you have any questions, please check out the video below. Happy calculating! 🙂

Circle Inscribed in a Triangle Construction:

2) We are going to create angle bisectors for each vertex of our triangle. Here we will start by making an arc with the point of the compass on angle A.

3) Next, we are going to go to the left where the arc we made and the triangle intersect and make another arc above our angle. Now bring the point of the compass to the right side where the arc we made and the triangle intersect and make another arc above our angle.

4) Using a ruler, connect angle A to the point of intersecting arcs, all the way to the other side of our triangle.

5) Now we are going to do the same thing, and find the angle bisector of angle B and angle C. For a review on how to find an angle bisector step by step, check out this link here.

6) Notice, the point of intersection where all three angle bisectors meet within the triangle is called the incenter.

7) Now we are going to create a perpendicular line segment by bringing the point of the compass to the incenter and creating an arc that intersects with triangle ABC.

8) Take our compass point to where the arc we just created and the triangle side intersect to create a new arc outside the triangle. Repeat on both sides of intersection.

9) Next, take a ruler align the incenter and intersection point we just created and draw a perpendicular line.

10) Using a compass, measure the length of the radius formed by the pink perpendicular line we just created. This will be the length of the radius of our circle!

11) Keeping that same distance we measured in the last step, draw a circle with the point of the compass at the incenter.

12) We have successfully inscribed a circle within our triangle ABC.

*Please note that this construction will work for different angles and for any type of triangle!

Constructions and Related Posts:

Looking to construct more than just a circle inside a triangle? Check out these related posts and step-by-step tutorials on geometry constructions below!

Construct an Equilateral Triangle

Perpendicular Line Segment through a Point

Angle Bisector

Construct a Square Inscribed in a Circle

Altitudes of a Triangle (Acute, Obtuse, Right)

How to Construct a Parallel Line

Bisect a Line Segment

How to Construct a 45 Degree Angle

Best Geometry Tools!

Looking to get the best construction tools? Any compass and straight-edge will do the trick, but personally, I prefer to use my favorite mini math toolbox from Staedler. Stadler has a geometry math set that comes with a mini ruler, compass, protractor, and eraser in a nice travel-sized pack that is perfect for students on the go and for keeping everything organized….did I mention it’s only $7.99 on Amazon?! This is the same set I use for every construction video in this post. Check out the link below and let me know what you think!

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Looking for more constructions? Check out how to construct a square inscribed in a circle and an equilateral triangle by clicking on their respective links! And if you’re looking for even more geometry constructions, check out the link here!

Still got questions? Check out the video above and comment with any questions below. Happy calculating! Also, be sure to check us out on social media for exclusive math tips and tricks!

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