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Inscribed And Circumscribed Circles

Have you ever wondered if you can fit a perfect circle inside a triangle, or draw one around the outside that touches all its corners? You can! This lesson explores inscribed circles (inside) and circumscribed circles (outside), showing you how to find their centers and understand their unique properties.

What Are Inscribed and Circumscribed Circles?

An inscribed circle is the largest possible circle that can be drawn inside a polygon, touching each of the polygon's sides at exactly one point. The polygon is said to be circumscribed about the circle. Think of it as a circle snugly fitted inside a shape, touching all the walls.

A circumscribed circle is a circle that passes through all the vertices (or corners) of a polygon, with the polygon lying completely inside the circle. In this case, the polygon is said to be inscribed in the circle. Imagine connecting the dots on a circle's edge to form a shape inside.

While these concepts can apply to many polygons, they are most commonly studied with triangles, because every triangle has both a unique inscribed circle and a unique circumscribed circle.

How Do You Find the Circumscribed Circle of a Triangle?

To draw the circumscribed circle for a triangle, you first need to find its center. This special point is called the circumcenter.

The circumcenter is the point where the perpendicular bisectors of a triangle's sides intersect. A perpendicular bisector is a line that does two things:

It passes through the midpoint of a side, cutting it into two equal halves.
It is perpendicular (at a $90^{\circ}$ angle) to that side.

Every triangle has three sides, so it has three perpendicular bisectors. Amazingly, they all meet at a single point—the circumcenter! The key property of the circumcenter is that it is equidistant from all three vertices of the triangle. This common distance is the radius of the circumscribed circle, often called the circumradius.

The location of the circumcenter depends on the type of triangle:

Acute Triangle: The circumcenter is inside the triangle.
Obtuse Triangle: The circumcenter is outside the triangle.
Right Triangle: The circumcenter is exactly at the midpoint of the hypotenuse (the longest side).

Worked Example: Constructing a Circumcenter

Example 1

Let's find the circumcenter of an acute triangle, $△ A B C$ . While we won't use coordinates, we can describe the geometric construction.

Step 1: Find the perpendicular bisector of side $\overset{―}{A B}$ .
First, find the midpoint of the segment $\overset{―}{A B}$ . Let's call it $M$ . Then, draw a line through $M$ that forms a $90^{\circ}$ angle with $\overset{―}{A B}$ . This line is the perpendicular bisector.

Step 2: Find the perpendicular bisector of side $\overset{―}{B C}$ .
Similarly, find the midpoint of $\overset{―}{B C}$ , which we'll call $N$ . Draw a line through $N$ that is perpendicular to $\overset{―}{B C}$ .

Step 3: Locate the point of intersection.
The two perpendicular bisectors you drew will cross at a single point. Let's call this point $P$ . This point $P$ is the circumcenter of $△ A B C$ . You could also draw the third perpendicular bisector (for side $\overset{―}{A C}$ ), and you would find that it also passes through point $P$ .

Step 4: Draw the circumscribed circle.
The distance from $P$ to vertex $A$ is the circumradius. Let's call it $R$ . The distances $P B$ and $P C$ are also equal to $R$ . You can now set a compass to this radius, place the point at $P$ , and draw a circle that passes perfectly through vertices $A$ , $B$ , and $C$ .

How Do You Find the Inscribed Circle of a Triangle?

Just like the circumscribed circle, the inscribed circle also has a special center point. This point is called the incenter.

The incenter is the point where the angle bisectors of a triangle's angles intersect. An angle bisector is a line or ray that divides an angle into two smaller, equal angles. For example, an angle bisector of a $60^{\circ}$ angle would split it into two $30^{\circ}$ angles.

A triangle has three angles, and therefore three angle bisectors. These three lines always meet at a single point inside the triangle—the incenter. The most important property of the incenter is that it is equidistant from all three sides of the triangle. This distance is the radius of the inscribed circle, known as the inradius.

Unlike the circumcenter, the incenter is always located inside the triangle, regardless of whether it is acute, obtuse, or right.

There is a useful formula that connects a triangle's area to its inradius. If a triangle has an area $K$ , an inradius $r$ , and a semi-perimeter $s$ (which is half the perimeter), then:

K = rs

The semi-perimeter $s$ is calculated as $s = \frac{a + b + c}{2}$ , where $a$ , $b$ , and $c$ are the lengths of the triangle's sides.

Worked Example: Finding the Inradius

Example 2

A triangle $△ X Y Z$ has side lengths of $x = 15$ cm, $y = 14$ cm, and $z = 13$ cm. Its area is known to be $84$ square cm. Find the radius of its inscribed circle (the inradius).

Step 1: Identify the given information.
We are given the side lengths $a = 15$ , $b = 14$ , $c = 13$ , and the area $K = 84$ .

Step 2: Calculate the semi-perimeter (s).
The semi-perimeter is half the perimeter. The perimeter is $15 + 14 + 13 = 42$ cm.
So, $s = \frac{42}{2} = 21$ cm.

Step 3: Use the inradius formula $K = r s$ .
We have $K = 84$ and $s = 21$ . We need to solve for $r$ .
$84 = r \times 21$

Step 4: Solve for $r$ .
To isolate $r$ , we divide both sides by $21$ .
$r = \frac{84}{21}$
$r = 4$

Answer: The radius of the inscribed circle (the inradius) is $4$ cm. This means the center of the circle is exactly $4$ cm away from each of the three sides.

Can Other Shapes Have Inscribed and Circumscribed Circles?

Yes, but not all of them! While every triangle has both, other polygons have stricter rules.

Regular Polygons: Any regular polygon (like a square, a regular pentagon, or a regular hexagon) has both an inscribed circle and a circumscribed circle. Their centers are the same point, which is the center of the polygon itself.

Quadrilaterals (4-sided shapes):

A quadrilateral has an inscribed circle if and only if the sums of its opposite sides are equal. For a quadrilateral with sides $a, b, c, d$ , this means $a + c = b + d$ . Such a shape is called a tangential quadrilateral.
A quadrilateral has a circumscribed circle if and only if its opposite angles add up to $180^{\circ}$ . Such a shape is called a cyclic quadrilateral. A rectangle is a great example, as all its angles are $90^{\circ}$ and $90^{\circ} + 90^{\circ} = 180^{\circ}$ .

Example 3

Consider a regular hexagon with a side length of $6$ inches. Let's find the radius of its circumscribed circle and its inscribed circle.

1. Circumscribed Circle Radius (Circumradius):
A regular hexagon can be divided into $6$ identical equilateral triangles, with the center of the hexagon being a common vertex. The sides of these equilateral triangles are equal to the side length of the hexagon. The distance from the center to any vertex of the hexagon is simply the side length of one of these small triangles.
Therefore, the circumradius $R$ is equal to the side length.
$R = 6 inches$

2. Inscribed Circle Radius (Inradius):
The radius of the inscribed circle is the perpendicular distance from the center to a side. This distance is also the height (or apothem) of one of the equilateral triangles. We can find this height by splitting an equilateral triangle in half, creating two $30 - 60 - 90$ right triangles. The hypotenuse is $6$ , the short leg is $3$ , and the height (long leg) is $3 \sqrt{3}$ .
Therefore, the inradius $r$ is:
$r = 3 \sqrt{3} \approx 5.2 inches$

What Are Some Common Mistakes to Avoid?

Mixing up Centers: The most common error is confusing the incenter and the circumcenter. Remember: INcenter comes from ANgle bisectors. Circumcenter comes from the perpendicular bisectors of the sides.
Circumcenter Location: Don't assume the circumcenter is always inside the triangle. For obtuse triangles, it's outside!
Radius vs. Distance: The circumradius is the distance from the center to a vertex. The inradius is the distance from the center to a side.
Assuming All Polygons Work: Remember that not every quadrilateral can have an inscribed or circumscribed circle. A long, thin rectangle, for example, cannot have an inscribed circle that touches all four sides.
Confusing Radius and Diameter: When a question asks for the circle's properties, double-check if it's asking for the radius or the diameter (which is $2 r$ ).

Quick Summary: Key Concepts

Use this table as a quick reference for the key differences between inscribed and circumscribed circles of a triangle.

Feature	Inscribed Circle	Circumscribed Circle
Location	Inside the polygon	Outside the polygon
Relation to Polygon	Touches each side at one point	Passes through each vertex
Center Point	Incenter	Circumcenter
How to Find Center	Intersection of the angle bisectors	Intersection of the perpendicular bisectors of the sides
Key Property of Center	Equidistant from the sides	Equidistant from the vertices
Radius Name	Inradius	Circumradius
Center Location	Always inside the triangle	Inside (acute), outside (obtuse), or on (right) the triangle

Frequently Asked Questions

What's the easiest way to remember the difference between incenter and circumcenter?

A good mnemonic is: the INcenter is where the ANgle bisectors meet (both have 'N'). The Circumcenter is related to the Circumference, which goes around the outside, and it is found using the sides (perpendicular bisectors).

Can a circle be inscribed in a rectangle that isn't a square?

No. For a quadrilateral to have an inscribed circle, the sums of opposite sides must be equal. In a rectangle with length 'l' and width 'w', this would require l+l = w+w, which means l=w. This is only true for a square.

Can a circle be circumscribed about a rectangle?

Yes, every rectangle can have a circumscribed circle. This is because its opposite angles are both 90 degrees, and their sum is 180 degrees, which is the condition for a quadrilateral to be cyclic (have a circumscribed circle).

Is the incenter ever the same point as the circumcenter?

Yes, but only in an equilateral triangle. In an equilateral triangle, the angle bisectors are the same lines as the perpendicular bisectors, so the incenter and circumcenter are the exact same point.

What is the radius of the inscribed circle called?

The radius of the inscribed circle is called the inradius. Similarly, the radius of the circumscribed circle is called the circumradius.

Why is the circumcenter of a right triangle on the hypotenuse?

This is a special property. Any angle inscribed in a semicircle is a right angle. Therefore, if a right angle (the 90-degree vertex) is on the circle, the side opposite it (the hypotenuse) must be a diameter of the circle. The center of the diameter is the circumcenter.

Does every polygon have an inscribed circle?

No, only certain types of polygons are guaranteed to have one. All triangles and all regular polygons have inscribed circles, but many irregular polygons, like most rectangles and parallelograms, do not.

How are these concepts used in the real world?

These geometric principles are used in design, engineering, and architecture. For example, designing a circular park inside a triangular city block (inscribed circle) or finding a location for a water tower that is equidistant from three towns (circumcenter).