how to make a nice 90 degree angle

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How to Construct a 90 Degrees Angle Using Compass and Ruler

Last Updated: December 15, 2022 Fact Checked

wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors. To create this article, 10 people, some anonymous, worked to edit and improve it over time. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 137,937 times. Learn more...

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.

At the End of a Line Segment

• The measure of the angle LMN is 90 O .

Through the Middle of a Line Segment

• 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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Things You'll Need

• Pen or pencil
• Ruler or another straightedge

You Might Also Like

• ↑ https://www.youtube.com/watch?v=2jEfhX6icow
• ↑ https://www.mathsteacher.com.au/year8/ch10_geomcons/05_angles/const.htm
• ↑ https://www.mathsisfun.com/geometry/construct-90degree.html
• ↑ https://www.bbc.co.uk/bitesize/guides/z9pfcwx/revision/4
• ↑ https://www.mathopenref.com/constangle90.html

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How to Lay Out a 90-Degree Angle With Just a Measuring Tape

If you were awake when your high school math teacher was explaining the Pythagorean theorem, that knowledge and a tape measure are all you need to lay out an accurate 90-degree angle. Even if you missed class that day, it's still easy to understand the technique, which comes in handy for laying out new structures as well as checking the square on existing ones.

Right-Angle Measurement Using the Pythagorean Theorem

The key to laying out a perfect 90-degree angle is to construct a right triangle, which always has one 90-degree angle. According to the Pythagorean theorem, the lengths of the sides of any right triangle (a, b and c) are related by the expression:

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a 2 + b 2 = c 2

Now suppose the length of side "a" is 3 units and that of side "b" is 4 units. If you plug those numbers into the equation and solve, you'll find the length of side "c" to be 5 units.

The 3-4-5 method works for any values of "a" and "b" as long as you can reduce them to a 3:4 ratio. For example, if "a" is 6 and "b" is 8, then "c" is 10, and if "a" is 33 and "b" is 44, then "c" is 55. This is good to know when you have to change units.

How to Use the 3-4-5 Rule

Suppose you want to construct a fence, and you've set the first corner post. You want to be sure the lines that extend from that post form a 90-degree angle at the post. Here's how to do it:

• Snap a chalk line or stretch a string in the direction of one side of the fence. Measure 3 feet along that line with a tape measure and make a mark.
• Create another line in the general direction of the other side of the fence and make a mark at the 4-foot point on that line.
• Extend the tape measure between the marks. Without changing its distance from the post, adjust the position of the second mark until it is exactly 5 feet distant from the first mark. The angle between the fence lines is now exactly 90 degrees.

If you don't have string or chalk line, you can still employ this method using only your tape measure. Just extend the tape out and make marks on the ground at the appropriate distances from the post.

Checking Square With the 3-4-5 Method

The Pythagorean theorem comes in handy when you're framing walls, hanging doors or building cabinets. One way to ensure the angle between two sides is 90 degrees is to check it with a framing square, but you can also mark 3 units on one side and 4 units on the other and then measure the distance between them to ensure it's 5 units.

Contractors use a variation of the 3-4-5 method to check the square of door openings. They measure the distance from one top corner to the diagonally opposite bottom corner and compare this to the opposite diagonal. Given that the two sides of the frame are the same height and the top and bottom of the frame are the same length, the diagonal distances should be the same.

If they aren't, the opening must be out of square, and that usually means one of the sides isn't plumb. To determine which side, measure 3 units along the top, 4 along each side, make marks and then measure the distance between the marks. The side that does not measure 5 units between the marks is the one that needs adjusting.

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90° Degree Angle

How to construct a 90° degree angle using just a compass and a straightedge

Constructing 90 Degrees Angle

The construction of an angle of 90 degree s can be done in two ways - by using a compass and by using a protractor. Making a rough sketch is different from that of construction in geometry. In geometry, by construction, we mean drawing shapes, lines, angles with more accurate measurements. The pieces of equipment that we would need are a ruler, compass, and a protractor . Construction in geometry always starts with a point. We then extend to draw a line segment or a line and from there on construct the required shape or angle. An angle in geometry is the space between two lines that intersect each other. 90° denotes a right angle in geometry. Sometimes, there is a need to construct angles without using a protractor. Let us learn how to construct an angle of 90 degrees in this article.

Constructing 90 Degree Angle Using Compass

In this section, you will learn how to construct a 90 degrees angle with the help of a compass. The steps to construct an angle of 90 degrees are as follows.

Step 2: Using the compass and with any width as radius and with A as a center, draw an arc that cuts the line segment AB at a point and label it as C.

Step 3: With the same radius and with point C as the center, draw an arc that cuts the previous arc and name it D.

Step 5: With the same radius and with D and E as centers, draw two arcs that cut each other at a point and label it as F.

Construction of 90 Degree Angle With Protractor

Constructing an angle of 90° can also be done using a protractor. A protractor is a geometric instrument that has markings of angles on it. Let us look into the steps to do the same.

Step 1: Draw a line segment XY of any measurement.

Step 2: Place the midpoint or center of the protractor on point X.

Step 3: Starting from 0° in the protractor, move in the anti-clockwise direction and mark a point in the place of 90° as Z.

Step 4: Join the points Z and X. Now, ∠ZXY measures 90°.

Related Articles

Check out some interesting articles related to constructing an angle of 90°. Click to know more!

• Angle Bisector
• Points and Lines
• Perpendicular Bisectors

Solved Examples on Constructing 90 Degrees Angle

Example 1: Construct a perpendicular bisector for a line segment of length AB = 8 units and check if the angle formed by the perpendicular bisector with the line segment is 90° or not.

Step 1: Draw a line segment with length AB = 8 units.

Step 2: With A as the center, and more than half the length of AB as radius, draw an arc on both the sides of the line segment AB. Repeat the same process with B as center and the same radius. Mark the points of intersection as P and Q.

Step 3: Join the points P and Q. PQ is the perpendicular bisector of AB. Also, we can observe that the perpendicular bisector makes an angle of 90° with the line segment AB.

Example 2: What are some real-life examples where we can observe 90°?

The angles formed by the arms of the windmill, the clock showing nine o'clock, and the angle formed at the corners of rectangular windows and doors are some real-life examples where we can observe 90°.

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Practice Questions on Constructing an Angle of 90 Degrees

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FAQs on Constructing an Angle of 90 Degrees

What is an angle of 90° called.

An angle of 90° is called a right angle . Constructing an angle of 90° can be done by measuring 90° in the protractor or by constructing a perpendicular bisector to a straight line. The perpendicular bisector makes 90° with a straight line.

How Do You Construct an Angle of 90 Degrees With a Compass and a Ruler?

To construct an angle of 90° with a compass and a ruler, follow the steps mentioned below:

• Draw a line and label its endpoints as A and B.
• Using the compass with any width as radius and with A as a center, draw an arc that cuts the line segment AB at a point and label it as C.
• With the same radius and with point C as the center, draw an arc that cuts the first arc, and name it as D.
• With D as a center and with the same radius draw another arc and name it as E.
• With the same radius and with D and E as center draw two arcs that cut each other and label it as F.
• Now join the lines F and A. Now we can observe that angle FAB = 90°.

How Do You Construct an Angle of 90 Degree With a Protractor and a Ruler?

To construct an angle of 90° with a protractor and a ruler, follow the steps mentioned below:

• Draw a line segment and label it as XY.
• Place the midpoint or center of the protractor on point X.
• Starting from 0° in the protractor, move in the anti-clockwise direction and mark a point in the place of 90° as Z.
• Join the points Z and X. Now the angle ZXY measures 90°.

What is the Angle Formed by a Perpendicular Bisector?

The angle formed by a perpendicular bisector is 90°. A perpendicular bisector is a line segment that divides a line segment into two parts that are equal and makes an angle of 90°.

How to Construct an Angle of 90° and Bisect It?

Draw a line segment AB. Place the protractor on point A and measure 90° and label it as C. Join CA. Now, with A as center draw an arc on CA and AB and mark it as E and F. With E and F as center draw two arcs that cut each other and label it as G. Now, join G and A. The line segment GA bisects 90° and forms two angles each of 45 degrees.

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In geometry, there are a few basic rules that govern the construction of angles. One of these rules is that it is possible to construct an angle of 90 degrees using only a compass and a straight edge. This can be done by following a few simple steps.

Step 1: Begin by placing the point of your compass at any point on the paper. Step 2: Next, use the compass to draw a small circle with that initial point as the center. Step 3: Now, without changing the width of the compass, place the point of the compass on another point along the circle and draw another small circle. Step 4: Repeat step 3 until you have drawn several concentric circles.

Step 5: Once you have drawn several circles, find two points where two of the circles intersect. Step 6: Place the point of your compass at one of those intersection points and draw a line through both intersection points. The line you have just drawn will be your angle bisector.

Step 7: Finally, measure the angle you have just constructed using a protractor. If it measures 90 degrees, then you have succeeded in constructing an angle of 90 degrees!

Constructing an angle of 90 degrees in geometry is a relatively simple process that can be completed using only a compass and a straight edge. By following the steps outlined above, anyone can do it!

How do you construct a 90-degree angle?

There are a few different ways to construct a 90-degree angle. One way is to use a compass and draw two arcs that intersect at90 degrees. Another way is to use a ruler and draw a line that is perpendicular to another line. Finally, you can use a protractor to measure an angle and make sure it is 90 degrees.

How do you construct an angle in geometry?

There are a few different ways to construct an angle in geometry. One way is to use a compass and draw two arcs that intersect at the desired angle. Another way is to use a ruler and draw a line that is perpendicular to another line at the desired angle. Finally, you can use a protractor to measure an angle and make sure it is the desired angle.

How do you draw a 90-degree angle without a protractor?

There are a few different ways to draw a 90-degree angle without a protractor. One way is to use a compass and draw two arcs that intersect at 90 degrees. Another way is to use a ruler and draw a line that is perpendicular to another line. Finally, you can mark off equal distances on a line and connect the marks to create a 90-degree angle.

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Constructing a 90° (right) angle

This is the step-by-step, printable version. If you PRINT this page, any ads will not be printed.

See also the animated version .

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How to Construct 30°, 60°, 90° and 120° Angles

Construction of angles

In geometry, everything neatly fits together. Knowing how to bisect a line segment means we know how to make a  90°  angle. Knowing an equilateral triangle has three  60°  angles makes an easy job of constructing a  60°  angle with only a compass, a pencil, a straightedge, and some paper.

How to construct a 90 degree angle

On the paper, draw a line segment longer than the ray of a the 90° angle you need.

Locate two points on the line segment at either end.

Open the drawing compass to extend a bit beyond half the distance of the line segment (do this visually; no need for numbers).

Swing an arc above and below the line segment.

Without changing the compass, relocate it to the other endpoint.

Swing another arc above and below the line segment. The two arcs should cross above and below the line segment.

Connecting the arc intersection points with the straightedge will produce a neat, sure  90°  angle, with your transversal and the original line meeting at four right angles!

How to construct a 60 degree angle

60° forms interior angles of equilateral triangles. It is exactly 1 6 \frac{1}{6} 6 1 ​ of a circle. Half of it plus itself forms a right angle. You could go on and on.

Steps of construction of a 60 degree angle using a compass

Making a  60°  angle begins by remembering that equilateral triangle.

Construct a line segment with a straightedge. Label its endpoints. In our drawing, we will call them Points O and G .

Place the drawing compass needle on Point O and adjust it to meet Point G .

Swing an arc upward from Point G high above the line segment.

Without adjusting the compass, relocate the needle to Point G .

Swing an arc upwards, so it crosses the first arc.

Use the straightedge to construct a line segment from Point O up to the point of intersection of the two arcs.

Label that line segment's new endpoint Point D .

The angle created from Points  D  to  O  to  G , by striking three congruent lengths, is  60° . If you wanted, you could connect Points  D  and  G  and form the equilateral triangle. Hey,  DOG , you did it!

How to construct a 30 degree angle

A  30°  angle is half of a  60° angle. So, to draw a  30° angle, construct a  60°  angle and then bisect it.

First, follow the steps above to construct your  60°  angle.

Bisect the  60°  angle with your drawing compass, like this:

Without changing the compass, relocate the needle arm to one of the points on the rays. Swing an arc on the inside of the angle.

Without changing the compass, relocate the needle arm to the other ray's point. Swing an arc on the inside of the angle.

Use the straightedge to connect the intersection of the two arcs with the vertex of the 60° angle. That line segment is an angle bisector, yielding two 30° angles.

How to construct a 120 degree angle using a compass

Remember how we said everything in geometry fits together? What is the supplementary angle to an angle of  120° ? In other words, what angle must we add to  120°  to get  180° ?

Did you say  60° ? Sure, a  120°  angle is the adjacent angle to any of the  60°  angles you already constructed!

To construct your  120°  angle, construct a  60°  angle and then extend one of its sides far past the vertex.

That angle beyond the  60°  angle is your  120°  angle.

None of your constructions required numbers or measurement. Euclid would be proud of you!

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Constructing a 90°, 45° or 22.5° Angle

Instructions for constructing 9 0 ° angles, constructing a 4 5 ° angle.

When you need to find an angle that is half of another angle, you always use the bisection technique . When you construct a 4 5 ° angle, you just bisect a 9 0 ° angle. You simply make a cross, midway between the two sides of the angle, with your draft compass, and draw a line from the vertex through this cross. The two new angles are both 4 5 ° , because 4 5 + 4 5 = 9 0 .

Constructing a 2 2 . 5 ° Angle

Once again, you need to find an angle that is half of another. For that reason, you can just use the bisection method one more time. So when you construct a 2 2 . 5 ° angle, you just bisect a 4 5 ° angle. You simply make a cross midway between the two sides of the angle with your draft compass, and draw a line from the vertex through this cross. The two new angles are both 2 2 . 5 ° , because 2 2 . 5 + 2 2 . 5 = 4 5 .

Construct a 90 degree angle Video

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How to Lay Out Right Angles in Construction

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Laying out accurate right angles on building projects — such as foundations for sheds, decks or patios — is easy if you use geometry.

According to the Pythagorean Theorem, the square of the two sides of a triangle that adjoin the right angle (legs) are equal to the square of the third side (hypotenuse). This is expressed mathematically as a² + b² = c².

To use, multiply the length of each leg of the triangle by itself then add the two sums together to find the length of the hypotenuse when the angle is at 90°.

The easiest way to accomplish this is to use the 3-4-5 method:

• Measure 3 feet out from the angle you want to make 90° in one direction.
• Measure 4 feet out from the angle you want to make 90° in the other direction.
• Measure across the two points and adjust the angle until the distance on the third side of the triangle is 5 feet.

You can also use multiples of 3-4-5 in the same ratio (such as 6, 8, 10) to form larger or smaller right angles.

Watch the video above to find out more.

VIDEO TRANSCRIPT

Joe Truini: You might not have thought you’d ever get around to using high school geometry, but if you’ve ever had to lay out lines at a perfectly square, 90-degree corner, here’s a chance to use it.

If it’s a large project like this, where we’re extending the patio, a framing square would be too small; it wouldn’t be accurate enough. So we’re going to use the Pythagorean theorem and that’s based on a three, four, five ratio.

So along one line, I measured and marked three feet, and along the intersecting line I did the same thing, only at four feet. And now, to put the theorem into practice, you simply measure across the two lines and you move the stake, in or out, until the five-foot mark lines up exactly with the mark that you made on the line. Then you can drive in the stake.

And it won’t be perfectly in line the first time, but you can move the stake in or out, side to side, as you need to.

And this works — here I did it at three feet, four feet, five feet. But you can use any of those ratios, so it could be six, eight or 10. The larger the project, the larger the numbers, the more accurate it’ll be.

Further Reading

• How to Find Square Layout Lines  
• How to Make Long Measurement Accurately  
• How to Make Tape Measure Reading Simple

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How to Calculate a 90-Degree Angle

How to Draw an Octagon or 8 sided Polygon

The 90-degree angle, also known as a right angle, is one of the most prevalent angles used in architecture. The 90-degree angle, formed by two lines that are perpendicular to each other, is a basic geometric concept. Geometric shapes such as squares and rectangles use right angles exclusively. There are a number of ways to create a 90-degree angle or determine whether an angle is 90 degrees, depending on the application, the tools and information at hand.

Measure the angle with a protractor if possible. If the angle in question is in a diagram or illustration, and you are sure that the drawing is to scale, then measurement with a protractor is a precise and accurate way to determine the degree of the angle. Simply line up the bottom of the protractor with the angle's adjacent side, align the angle's point with the protractor's cross hairs and note the measurement marking that the opposite side indicates.

Use mathematical inference to determine the angle. If the angle you are analyzing isn't in a diagram that's drawn to scale, or if the diagram is too small to use a protractor, you can often use basic geometric principles to determine the measure of an angle. If the angle is indicated with a small square rather than a curved line, then the angle is meant to be 90 degrees. If the angle is a part of a shape such as a rectangle or square, then the angle is 90 degrees. If the angle is part of a triangle, add the measures of the other two angles. The three angles of a triangle always total 180 degrees, so if the sum of the other two angles is 90, then it is a right triangle with a 90-degree angle. Similarly, the four angles of a quadrilateral (a shape with four sides) always total 360. So, if adding the other three angles yields 270, then the angle in question is a right angle. There are a number of other geometric theorems and postulates that can be used to infer the measure of an angle.

Measure the angle with a special tool. If you are trying to measure the angle of a cut of wood, for example, you can use a T-square or other measuring square. If you are trying to measure the angle of a star or some similar celestial figure, you can use a sextant, angle finder or some similar line-of-sight-based measuring tool. In addition, a spirit level, also known as a bubble level, can be used. If the bubble is in between the two parallel lines, then the level is at a 90-degree angle from horizontal. Conversely, if the level is laid flat or held sideways and adjusted until the bubble is in position, then the level is at a 90-degree angle from vertical.

Calculate whether an angle is 90 degrees using the Pythagorean Theorem. This well-known theorem is often phrased as "A squared plus B squared equals C squared," which indicates that the sum of the squares of lengths of the adjacent sides of a right triangle is equal to the square of the length of the hypotenuse side. With a bit of mathematical ability, this formula (a^2 + b^2 = c^2) can be manipulated and used to determine a right angle. Using a ruler, measure the sides of the angle as well as the distance between the angle's open endpoints. If these values plug into the formula correctly, then the angle is a 90-degree angle.

Things You'll Need

• A basic understanding of geometric theorems and principles will help you to determine unknown angles in technical drawings and other geometric figures.

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• Regents Exam Prep Center: Theorems and Postulates for Geometry
• Rickler: Marking and Measuring Tool Basics

About the Author

Crispin Trubiano has been working as a freelance writer since 2010. His articles appear on various websites, where he specializes in areas such as technology, health, television, film, literature and music. Trubiano currently studies sociology at Roger Williams University.

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