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### Course: High school geometry > Unit 8

Lesson 11: Constructing regular polygons inscribed in circles# Geometric constructions: circle-inscribed square

Sal constructs a square that is inscribed inside a given circle using compass and straightedge. Created by Sal Khan.

## Want to join the conversation?

- how does this construction prove that this square is actually a square(20 votes)
- It begins with a diameter, then constructs the perpendicular bisector of the diameter (by drawing the line through the lemon formed by the overlapping congruent circles placed at the endpoints). These form the diagonals of your quadrilateral.

(1) They bisect each other, because they cross at the center.

(2) They are congruent, since they are both diameters.

(3) They are perpendicular, per construction.

(1) proves it is a parallelogram, (2) that it is a rectangle, and (3) that it is a rhombus. Thus it is proven to be a square.(39 votes)

- difference between polygon and regular polygon(9 votes)
- A polygon is any type of closed figure in which all sides are straight. A regular polygon is a polygon in which all sides are the same length and all angles have the same measure.(20 votes)

- How did Sal decide how big the blue circles were at1:00to1:10(5 votes)
- The first circle needs to be at least half the diameter of the circle win the problem. Then, with the second circle, Sal simply placed the midpoint of the new compass on the midpoint of the first circle, made the circles the same size by adjusting the new compass, and then placed the new circle's midpoint on the opposite side of the circle in the problem.

Hope this helps :)(2 votes)

- In which website in Khan Academy can i visit to move the compass and straightedge where? does someone know?(3 votes)
- There might be one, but for all I know, It could have been removed.(1 vote)

- Why doesn't the site have construction exercises anymore?(3 votes)
- how do we know its an actual square(2 votes)
- Since it is a perpendicular bisectors of equal length that are diameters, each part forms a radius. Since the 90 degree angle is there and the sides are equal, the other angles have to be 45 degress, and each of the 4 triangles have to be similar, so you have angles of 90 degrees total (rectangle requirements) and with perpendicular bisectors and all sides equal (rhombus requirements), it has to be a square.(2 votes)

- which website did you use?(2 votes)
- Sal completed the exercise on the Khan Academy website.(1 vote)

- In a make up of a circle I notice there are six pentagons within hexagons, is it the curved shap of the circle itself and its angle ? How do you get the five sides of the pentagon to line up with the six sides of the Hexagon?(2 votes)
- What is a four sided figure called in which one side is straight ?(1 vote)
- can you inscribe a rectangle in a circle?(2 votes)
- You can inscribe a rectangle in a circle.

Technically, a square IS a rectangle with all sides equal.

So, yes even this tutorial teaches you how to inscribe a rectangle in a circle ,too!(1 vote)

## Video transcript

Construct a square
inscribed inside the circle. And in order to do
this, we just have to remember that a square,
what we know of a square is all four sides are
congruent and they intersect at right angles. And we also have to
remember that the two diagonals of the
square are going to be perpendicular
bisectors of each other. So let's see if we can
construct two lines that are perpendicular
bisectors of each other. And essentially, where those
two lines intersect our bigger circle, those are going to be
the vertices of our square. So let's throw a straight
edge right over here. And let's make a diameter. So that's a diameter
right over here. It just goes through
the circle, goes through the center
of the circle, to two sides of the circle. And now, let's
think about how we can construct a perpendicular
bisector of this. And we've done this in
other compass construction or construction videos. But what we can do is we
can put a circle-- let's throw a circle right over here. We've got to make its radius
bigger than the center. And what we're going to do
is we're going to reuse this. We're going to make
another circle that's the exact same size. Put it there. And where they intersect
is going to be exactly along-- those two
points of intersection are going to be along a
perpendicular bisector. So that's one of them. Let's do another one. I want a circle of the
exact same dimensions. So I'll center it
at the same place. I'll drag it out there. That looks pretty good. I'll move it on to this side,
the other side of my diameter. So that looks pretty good. And notice, if I connect
that point to that point, I will have constructed
a perpendicular bisector of this original segment. So let's do that. Let's connect those two points. So that point and that point. And then, we could just
keep going all the way to the end of the circle. Go all the way over there. That looks pretty good. And now, we just have
to connect these four points to have a square. So let's do that. So I'll connect
to that and that. And then I will connect, throw
another straight edge there. I will connect that with that. And then, two more to go. I'll connect this with
that, and then one more. I can connect this with
that, and there you go. I have a shape whose vertices
intersect the circle. And its diagonals, this diagonal
and this diagonal, these are perpendicular bisectors.