Inscribed and Circumscribed Figures: Definition & Construction

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  • 0:05 At the Pentagon
  • 0:34 Inscribed and…
  • 1:46 Inscribed Polygons
  • 3:08 Inscribed Circles
  • 3:46 Practice Problems
  • 5:45 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

A square and a circle may be different shapes, but they still can have a unique relationship. In this lesson, we'll learn about inscribed and circumscribed figures.

At the Pentagon

Meet Emma. Emma works at the Pentagon. That's right, everyone's favorite five-sided building. Well, maybe you don't love the Pentagon. That's OK. But you have to admire its, well, five-sided-ness. It's definitely a pentagon.

Emma's job involves running meetings in all different parts of the Pentagon. What does that mean? It means she has to get from her office here to meetings all along this path.

Inscribed Circle
Circle Inscribed

Inscribed and Circumscribed Figures

What is this path? It's a circle. Since it's inside a pentagon like this, it's called an inscribed figure. An inscribed figure is a shape drawn inside another figure. We say that the circle is inscribed in the pentagon.

When she has a break, Emma likes to go for a run. She usually doesn't have much time, so she runs the shortest possible route around the building. Her path looks like this. Now our circle is outside the pentagon, and now it's not inscribed but circumscribed. A circumscribed figure is a shape drawn outside another figure.

Circumscribed Circle
Circle Circumscribed

Wait. What's the difference? Couldn't you say that the pentagon is inscribed in the circle, rather than the circle is circumscribed on the pentagon? Yes! The difference between inscribed and circumscribed simply is a matter of which figure is being described in terms of the other.

Remember that the 'in'-scribed figure is 'in'-side the other. And the 'circ'-umscribed figure 'circ'-les the other. That's true even here, when it's the pentagon that circumscribes the circle.

Inscribed Polygons

Eventually, Emma moves on from the Pentagon and gets a job at the Capitol. Folks in Congress like to have impromptu meetings under the rotunda. So Emma is often moving around in this circular-shaped area.

Inscribed Triangle inside a circle
Triangle Inscribed

Each meeting forms a corner on our shape. And each corner is also known as a vertex. When we have more than one vertex, we say vertices. Seems fancy, right? Well, this is Congress.

Here's a triangle showing the path Emma takes for three meetings with senators who don't like to be near each other. That's important, since for a shape to be inscribed in a circle, all of its vertices must be on the circle.

If the senators were more friendly, and the triangle looked like this, we wouldn't say that the triangle is inscribed on the circle. The vertices don't touch the circle.

Triangle not inscribed
Triangle not inscribed

And not every shape can be inscribed in a circle. Triangles and pentagons can. Here's a square. And here's a rectangle. Those are both parallelograms that work. What about any parallelogram? Not this one. Only two sides touch the circle. So not every parallelogram can be inscribed, just a square or rectangle. With any shape, just check to be sure all the vertices touch the circle.

Four sides inscribed
Inscribed four sides

Not inscribed parallelogram
Not inscribed four sides

Inscribed Circles

When Emma is on the floor of the House, which is a square, she moves in a circular pattern, dodging representatives.

When we're talking about circles, we look for the circle to be tangent to the sides of the shape. With our square here, note that the circle is tangent to all four sides of the square. Tangent just means that it touches in only one place.

Just as before, this will only work with some shapes. Take the triangle. Some circles can be inscribed in some triangles, like you see here. But what about here? The circle doesn't touch every side, so nope.

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