Back To CourseGeometry: High School
15 chapters | 160 lessons
As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.Try it risk-free
Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.
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.
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.
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.
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.
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.
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.
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.
But when the circle is inscribed, we can do some neat things. Let's say we want to know the area of the shaded region here. We know the radius of the inscribed circle is 4. What's the area of the circle? pi(r)^2. So 16pi. What about the entire square? Well, if the radius of the circle is 4, and the circle touches all sides of the square as it does, then the side of the square is 8. So its area is 8^2, or 64. That means the shaded area is 64 - 16pi. That rounds to about 14.
This works with inscribed polygons as well. Here's a square inscribed in a circle. Notice that all the vertices touch the circle. If the radius of the circle is 9, what is the area of the shaded region? Well the area of the entire circle is pi(r^2), or 81pi.
What about that pesky square? If the radius of the circle is 9, then the diagonal of the square is 18. Did you know that you can find the area with just this number? Remember, a square is just two right triangles put together. So our triangle base is 18. And the height of the triangle? It's the radius of the circle, or 9. That means that the area of each triangle is 1/2bh, or 1/2*18*9, which is 81. That makes the entire area of the square 162. Incidentally, I just did a quick version of the proof, but the formula is just d^2/2, where d is the diagonal. And 18^2/2? That's 324/2, or, yep, 162. So the area of the shaded region is 81pi - 162. That's about 92.
In summary, an inscribed figure is a shape drawn inside another shape. A circumscribed figure is a shape drawn outside another shape.
For a polygon to be inscribed inside a circle, all of its corners, also known as vertices, must touch the circle. If any vertex fails to touch the circle, then it's not an inscribed shape.
For a circle to be inscribed inside a polygon, it must be tangent to, or touch, all sides of the shape.
As for Emma, all of this moving around is pretty exhausting. Sometimes she feels like she's trying to fit a square peg in a round hole. Oh, inscribed shape humor...
Watch this video in order to:
To unlock this lesson you must be a Study.com Member.
Create your account
Already a member? Log InBack
Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.
To learn more, visit our Earning Credit Page
Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.
Back To CourseGeometry: High School
15 chapters | 160 lessons