How to Construct a 90 Degrees Angle Using Compass and Ruler

Co-authored by Mario Banuelos, PhD

Last Updated: December 15, 2022 Fact Checked

Often times, you are required to construct some angles without using a protractor. This article teaches you how to draw a 90 degrees angle using a compass and a ruler.

Method 1

Method 1 of 2:

At the End of a Line Segment

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Let us name this point as M.
This will be one of the arms of our angle.^{[1]
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^{[2]
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Then place its tip on P and draw an arc cutting the circle at some point (say Q).
Place its tip on Q and draw another arc cutting the circle at another point (say R).^{[3]
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Draw another arc somewhere outside the circle. For carrying out this step, you can set the width of the compass to any measure.^{[4]
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Now place its tip on R and draw another arc which cuts the arc drawn in the previous step at some point (say S).
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Connect the points M and S with a straight line. Extend it to form a ray ML.^{[5]
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- The measure of the angle LMN is 90^O.

Method 2

Method 2 of 2:

Through the Middle of a Line Segment

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On a piece of paper, draw a line segment of any convenient length. Make it horizontal for the sake of simplicity, but it could be at any angle. Label the line segment AB.^{[6]
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Mark it, and label it O. O can be anywhere on the line segment, including at either end point (A or B). For convenience, let's put O somewhere near the middle of AB. The line segment could now be referred to as AOB.
Open the compass to a radius a little shorter than either AO or OB. Now place the compass point at O. Strike short arcs through AB on either side of O. The two arcs should have the same radius. Label those two points of intersection C and D. Now the line segment could be referred to as ACODB.^{[7]
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Place the compass point at C, and strike an arc to one side of the line segment. Then place the compass point at D, and strike a second arc of the same radius and on the same side of the line segment. Make sure those two arcs intersect. Call that point of intersection E.^{[8]
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Draw the 90° angle. Draw a straight line from E to O. Line segment EO forms a 90° angle with line segment AB. It actually forms two 90° angles at O. If you were to extend EO beyond AB, you would form four 90° angles at O.^{[9]
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- Note that you can draw a 90° angle at either end of line segment AB if you want to (in other words at point A or point B). Simply extend AB beyond A or beyond B, and then follow the above steps. Point A (or point B) would serve as point O in the above instructions.
- This is essentially the same method featured in How to Construct a Perpendicular Line to a Given Line Through Point on the Line, Method 2.

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How do I draw a right angle triangle?

Donagan

Top Answerer

First construct a 90° angle as shown above. Extend each leg of the angle any convenient length. Draw a line from the end of one leg to the end of the other. You have a right triangle.

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How do I construct a 15 degree angle?

Community Answer

First, make a 60 degree angle by constructing an equilateral triangle. Use angle bisection construction to make a 30 degree angle. Bisect the 30 degree angle again to make a 15 degree angle.

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Can I draw a 133-degree angle?

Donagan

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There is no way to construct such an angle using a compass and a ruler. That's why they invented the protractor.

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