1. Why Drawing Parallel Lines Takes Two Set Squares
In elementary school math classes in Japan, students are taught to use two set squares to draw parallel lines. One square is held firmly in place, while the other is slid along it. [6] The Musashino Art University glossary also describes this method: fix one piece securely and move the other in a set direction to draw the line. [2]
The reason for using two pieces is mechanical. When you slide one set square along the edge of the fixed one, the moving square does not change its orientation. Therefore, the line drawn before the move and the line drawn after the move are parallel. If you only had one set square, there would be no fixed edge to guide the movement, so the angle might shift. [2]
In professional drafting, set squares are often used with a T-square. Placing a set square on top of a T-square and sliding it left or right allows the user to draw vertical lines or lines at fixed angles. [4] Many set squares have a hole in the center. This is explained as a way to prevent air from getting trapped between the tool and the paper, making it easier to keep the set square fixed in place. [3]
2. The Two Set Squares Are Half of a Square and Half of a Triangle
A standard set of set squares consists of two right-angled triangles. One has angles of 90°, 45°, and 45°. The other has angles of 90°, 60°, and 30°. [1]
The 45°-45°-90° triangle is the shape you get when you cut a square in half along its diagonal. The 30°-60°-90° triangle is what you get when you cut an equilateral triangle in half from a vertex down to the base. This shape is sometimes called a "half-equilateral triangle." [5]
The ratio of the side lengths differs for each shape. For the 45°-45°-90° triangle, the ratio is 1 : 1 : √2. For the 30°-60°-90° triangle, the ratio is 1 : √3 : 2. [3] There is no documentation found in this research that explains why these two specific shapes were selected for the standard pair. It is best to view these facts as the current standard form. [1]
3. The Long Sides of the Two Set Squares Are the Same Length
There is another key feature of this pair: the length of certain sides matches. The hypotenuse (the longest side) of the 45°-45°-90° triangle is equal in length to the longer leg (the side between the 90° and 30° angles) of the 30°-60°-90° triangle. According to Wikipedia, this shared length serves as a rough guide for the size of a set square. [3]
It is important not to confuse this with the hypotenuse of the 30°-60°-90° triangle, which is even longer. [3] If we assume this shared length is 30 cm, we can calculate the other sides using the ratios. For the 45°-45°-90° triangle, the two shorter sides are approximately 21.2 cm (30 ÷ √2). For the 30°-60°-90° triangle, the shortest side is approximately 17.3 cm (30 ÷ √3), and its hypotenuse is approximately 34.6 cm. [3]
Drafting-grade 45°-45°-90° set squares often have hypotenuses between 24 cm and 30 cm. [3] Because these lengths match, you can overlap the equal-length sides of the two set squares. But what does this alignment allow you to do? [3]
4. How Adding and Subtracting Angles Makes 15° and 75°
When you combine the two set squares, you can draw diagonal lines using angles of 30°, 45°, 60°, and 90°, in addition to horizontal and vertical lines. [1] By calculating the sum or difference of the angles on the two pieces, new angles emerge. For example, 45° minus 30° equals 15°, and 45° plus 30° equals 75°. Other combinations include 60° plus 45° for 105°, and 60° minus 45° for 15°. [2]
The English Wikipedia entry on set squares confirms that aligning the hypotenuses of the two pieces can create 15° and 75° angles. [4] However, this is a calculation of angles. The actual lines you can draw depend on which edges you align and how you position the tools. You cannot simply assume that any angle calculated on paper can be drawn easily in practice. [2]
5. Which Angles Need Other Tools and How Set Squares Were First Used
In the early Meiji period in Japan, these tools were referred to in documents as sankaku-ki. At that time, they were primarily used to check right angles. Angles other than 90° varied from tool to tool. [1] The glossary does not describe the history of how they became the standard two-piece set we know today. [1]
To draw lines at any arbitrary angle, there are other tools. A "bevel ruler" (or kōbai-jigoku) is a tool with a protractor and a scale plate, connecting two movable triangular pieces. While set squares can create various angles through combination, a bevel ruler allows you to draw lines at any arbitrary angle. [1]
In German-speaking schools, a tool called the Geodreieck is used. It combines a 45°-45°-90° triangle with ruler and protractor markings. It was developed by a German manufacturer in 1964. This is a different system from the Japanese set square pair and is not part of its origin story. [4]
6. How to Check Parallel Lines and Lengths with Set Squares at Home
If you have two set squares at home, you can try this in your notebook. Draw three parallel lines by holding one set square and sliding the other. Use a ruler to measure the space between the lines to check if the width is the same everywhere. [6]
Next, measure the long side of the 45°-45°-90° triangle and the side between the 90° and 30° angles on the other triangle. Check if they are close to the same length. Note that exact dimensions may vary slightly depending on the product. [3]
If you want to learn more, the Musashino Art University "Zokei File" glossary and their explanatory PDF on set squares include information on types like the bevel ruler and how to use them. These can be accessed through the reference links. [1] [2]