How to Construct an Isosceles Triangle

An isosceles triangle is a triangle with two equal side lengths and two equal angles. Sometimes you will need to draw an isosceles triangle given limited information. If you know the side lengths, base, and altitude, it is possible to do this with just a ruler and compass (or just a compass, if you are given line segments). Using a protractor, you can use information about angles to draw an isosceles triangle.

Method 1

Method 1 of 4:

Given All Side Lengths

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1
Assess what you know. To use this method, you should know the length of the triangle’s base and the length of the two equal sides. You can also use this method if you are given line segments representing the base and sides instead of the measurements.
- For example, you might know that the base of a triangle is 8 cm, and its two equal sides are 6 cm, or you might be given two lines, one representing the base, and one representing the two sides.
2
Draw the base. Use a ruler to make sure that your line is measured exactly. For example, if you know that the base is 8 cm long, use a sharp pencil and a ruler to draw a line exactly 8 cm long.
- If using a given line segment instead of a measurement, draw the base by setting the compass to the width of the provided base. Make an endpoint, then use the compass to draw the other endpoint. Connect the endpoints using a straightedge.
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3
Set the compass. To do this, open the compass to the width of the equal side lengths. If you are given the measurement, use a ruler. If you are given a line segment, set the compass so that it spans the length of the line.
- For example, if the side lengths are 6 cm, open the compass to this length. Or, if provided a line segment, set the compass to the segment's length.
4
Draw an arc above the base. To do this, place the tip of the compass on one of the base’s endpoints. Sweep the compass in the space above the base, drawing an arc.
- Make sure the arc passes at least halfway across the base.
Without changing the width of the compass, place the tip on the other endpoint of the base. Draw an arc that intersects the first one.
Use a ruler to draw lines connecting the point where the arcs intersect to either endpoint of the base. The resulting figure is an isosceles triangle.

Method 2

Method 2 of 4:

Given Two Equal Sides and the Angle Between Them

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1
Assess what you know. To use this method, you need to know the length of the two equal sides, and the measurement of the angle between these two sides. You can also use this method if you are given a line segment representing the side length instead of the measurement.
- For example, you might know that the isosceles triangle has two equal sides of 7 cm, or you might be given a line segment representing the side length. You also know that the angle between the sides is 50 degrees.
2
Draw the angle. Use a protractor to construct the angle of the given measurement. Ensure that each of its vectors is longer than the given side length.
- For example, you might need to draw a 50-degree angle. Since the sides of the triangle are 7 cm, the vectors should be a little longer than 7 cm long. You can use a ruler or your compass set to the appropriate length to measure.
3
Set the compass. If you know the measurement of the side lengths, use a ruler to open the compass to that length. If you are given a line segment instead of a measurement, use it to set the compass to the appropriate width.
- For example, if you know that the side lengths are 7 cm, then use a ruler to open your compass 7 cm wide.
To do this, place the tip of the compass on the vertex of the angle (where the two vectors meet). Draw one long arc that intersects each vector of the angle. You can also draw two small arcs, each one intersecting one of the vectors.
Using a straightedge, draw a line connecting the points where the arc intersects the two vectors. The resulting figure is an isosceles triangle.

Method 3

Method 3 of 4:

Given The Base and Adjacent Angles

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1
Assess what you know. To use this method, you need to know the length of the base, or you need to be provided with a line segment that represents the base. You also need to know the measurement of the two angles adjacent to the base. Remember that the two angles adjacent to the base of an isosceles triangle will be equal.^{[1]XResearch source}
- For example, you might know that an isosceles triangle has a base measuring 9 cm, with two adjacent 45-degree angles.
2
Draw the base. If you know the measurement of the base, use a ruler to draw it the appropriate length. Make sure to measure exactly, and to create a straight line.^{[2]XResearch source}
- You can also draw the base by setting the compass to the same width as a provided line segment. Draw an endpoint. Make the other endpoint using the compass. Then use a straightedge to connect the two endpoints.
Use a protractor to draw the angle on the left side of the base. The vector should pass a little more than halfway over the base, so that it will intersect with the other side of the triangle.^{[3]XResearch source}
Use a protractor to draw the angle on the right side of the base. Make sure the second vector intersects the first. Where the two lines intersect creates the apex of the triangle.^{[4]XResearch source} The resulting figure is an isosceles triangle.^{[5]XResearch source}

Method 4

Method 4 of 4:

Given the Base and Altitude

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1
Assess what you know. To use this method, you need to know the length of the triangle’s base, and the height, or altitude, of the triangle. You can also use this method if you are given line segments representing the base and altitude instead of the measurements.
- For example, you might have an isosceles triangle with a base of 5 cm and a height of 2.5 cm.
2
Draw the base. If you know the measurement, use a ruler. For example, if you know that the base is 5 cm long, use a ruler to draw a line that is exactly 5 cm long.
- If using a line segment instead of a measurement, draw the base by setting the compass to the width of the base. Draw an endpoint. Use the compass to draw the second endpoint. Then, connect the endpoints using a straightedge.^{[6]XResearch source}
3
Draw a line bisecting the base. This means a line that cuts the line in half. You can use a compass and the method described here. Draw the line at least as long as the triangle’s altitude.
- You can also use a ruler and a protractor to bisect the line. Divide the length of the base in half. Use the ruler to draw a midpoint. Then, use a protractor to draw a line at this midpoint that intersects the base at a 90-degree angle.
If you know the measurement of the altitude, use a ruler to open the compass to this exact length (for example, 2.5 cm). If you are given a line segment, open the compass to the length of the provided line.
Place the tip of the compass on the midpoint of the base. Draw an arc across the bisecting line. You need to draw the arc only on one side of the base.
Connect the point where the altitude and arc intersect with either endpoint of the base. The resulting figure will be an isosceles triangle.^{[7]XResearch source}

Community Q&A

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Question
How would you construct an isosceles right triangle if only given the hypotenuse?
Community Answer
If you know the length of the hypotenuse, you can find the length of the other two sides of the triangle using the Pythagorean theorem (a^2 + b^2 = c^2). However, since this is an isosceles triangle, the two sides will be the same length, so you will simplify the Pythagorean formula to x^2 + x^2 = c^2, or 2x^2 = c^2. For example, if the hypotenuse is 12 cm, the formula will be 2x^2 = 12^2:2x^2 = 12^22x^2 = 1442x^2/2 = 144/2x^2 = 72sqrt*x^2 = sqrt*72x = 8.48.Since every triangle has 180 degrees, if it is a right triangle, the angle measurements are 90-45-45. So the triangle will have a hypotenuse of 12, two side lengths of about 8.5 cm, and two 45 degree angles.
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Question
How do I construct a right isosceles triangle given perimeter?
Donagan
Top Answerer
You don't have enough information to do that.
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Question
If the base is 60 and the base angle is 45, what is the length of the two sides?
Donagan
Top Answerer
Both base angles are 45°. Therefore the third angle is 90°. Drop an altitude from the 90° angle to the base. The altitude bisects the 90° angle. It also bisects the base and is perpendicular to it. The altitude forms two smaller isosceles right triangles, each of which has two 45° angles and two sides with lengths of 30 (half the base). Thus, each 45° angle in each smaller right triangle has an opposite side and an adjacent side of length 30 and a hypotenuse of x (the length you're trying to find). The sine (and cosine) of each 45° angle is 0.707. Therefore, 0.707 = 30 / x. x = 30 / 0.707 = 42.4. That's the length of each of the two equal sides of the big triangle.
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