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## High school geometry

### Course: High school geometry>Unit 8

Lesson 11: Constructing regular polygons inscribed in circles

# Geometric constructions: circle-inscribed equilateral triangle

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

## Want to join the conversation?

• Why did Sal go to the trouble of constructing the initial small equilateral triangle before he connected the points where the circle he created intersects with the original circle? That line is (as he shows later) one leg of the larger equilateral triangle he wants to construct.
• he wanted to show why and how it works, not just tell you what to do. This should help you remember it, or at least figure it out yourself if you forget.
• at 203 i think Sal means...'a third of the way around the circle'. but he says triangle. 120 degrees is a third of a circle.
• No, Sal was correct there. A triangle turns 360 degrees from start to finish just like a circle does (or any closed polygon does). That side was 1/3 of the triangle, and just needed to be duplicated 2 more times.
• At . Why does Sal say, " third of the way around the triangle ". What does it mean?
Thanks.
• A triangle's three angles measure up to 180. Since it is a equilateral triangle Sal is constructing,
each angle is 60 degrees. Many people become confused when Sal says 120 degrees is one third of the triangle, but he is talking about the arc of the circle. To find the arc or the angle formed by the arc, use this equation: arcX = 2 angleX. Thus, the angle is 60 degrees and one third of a triangle, due to what I said earlier. (60 x 3 = 180)
• What does it mean to be a secant?
• A line that touches a circle in only two points. So basically a chord that extends infinitely.
• are there easier ways to solve these questions? it takes sal a long time to solve them
(1 vote)
• It's because Sal is explaining them in detail so that people can understand it better.
• Well cant he use this method :
The circumcentre of the triangle we will be going to make will be the center of the circle we are given. So we can make a diameter and treat it like the median of the triangle. Now all the median will intersect at circumcentre and hence the center. Now the intersecting point divides the median in 2:1. So we will be taking 2 as radius and perpendicularly bisect the other one to get 1/2 of radius or 1/2*2=1 and finally the ratio of 2:1.
Now the points of perpendicular bisector intersecting the circle can be connected to the first point of diameter. And its do equal to an equilateral triangle. If you are not understanding this then search for topics circumcircle and medians of equilateral triangle and please answer to this question.
• I get it but how do you do this on paper?
• At , how does Sal get the right place to put the compass?
(1 vote)
• Sal drags the compass so that its center is on the circumference of the original circle and he places it so that it "intersects these two points". The two points are the center of the original circle and the point where one side of the equilateral triangle that is already placed touches the circumference of the original circle.
• at between and , what does sal mean by either around the triangle or around the circle
• He meant third around the circle.
(1 vote)
• Could you make any shape in a circle?
• Technically speaking you could draw any you want inside a circle.
(1 vote)

## Video transcript

Construct an equilateral triangle inscribed inside the circle. So let me construct a circle that has the exact same dimensions as our original circle. Looks pretty good. And now, let me move this center, so it sits on our original circle. So they now sit on each other. Or their centers now sit on each other. So I can make it, and that looks pretty good. And now, let's think about something. If I were to draw this segment right over here, this, of course, has the length of the radius. Now, let's do another one. And that's either of their radii, because they have the same length. Now, let's just center this at our new circle and take it out here. Now, this is equal to the radius of the new circle, which is the same as the radius of the old circle. It's going to be the same as this length here. So these two segments have the same length. Now, if I were to connect that point to that point, this is a radius of our original circle. And so it's going to have the same length as these two. So this right over here, I have constructed an equilateral triangle. Now, why is this at all useful? Well, we know that the angles in an equilateral triangle are 60 degrees. So we know that this angle right over here is 60 degrees. Now, why is this being 60 degrees interesting? Well, imagine if we constructed another triangle out here, just symmetrically, but kind of flipped down just like this. Well, the same argument, this angle right over here between these two edges, this is also going to be 60 degrees. So this entire interior angle, if we add those two up, are going to be 120 degrees. Now, why is that interesting? Well, if this interior angle is 120 degrees, then that means that this arc right over here is 120 degrees. Or it's a third of the way around the triangle. This right over here is a third of the way around the triangle. Since that's a third of the way around the triangle, if I were to connect these two dots, that is going to be, this right over here is going to be a side of our equilateral triangle. This right over here, it's secant to an arc that is 1/3 of the entire circle. And now, I can keep doing this. Let's move-- I'll reuse these-- let's move our circle around. And so now, I'm going to move my circle along the circle. And once again, I just want to intersect these two points. And so now, let's see, I could take one of these, take it there, take it there, same exact argument. This angle that I haven't fully drawn, or this arc you could say, is 120 degrees. So this is going to be one side of our equilateral triangle. It's secant to an arc of 120 degrees. So let's move this around again. Actually, we don't even have to move this around anymore. We could just connect those last two dots. So we could just connect this one. Actually, I just want to, let's connect that one to that one. And just like that, and we're done. We have constructed our equilateral triangle.