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Set square
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Set square
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triangle, set square

A set square or triangle (American English) is an object used in engineering and technical drawing, with the aim of providing a straightedge at a right angle or other particular planar angle to a baseline.

Types

[edit]

The simplest form of set square is a triangular piece of transparent plastic (or formerly of polished wood) with the centre removed. More commonly the set square bears the markings of a ruler and a half circle protractor. The outer edges are typically bevelled. These set squares come in two usual forms, both right triangles: one with 90-45-45 degree angles, the other with 30-60-90 degree angles. Combining the two forms by placing the hypotenuses together will also yield 15° and 75° angles. They are often purchased in packs with protractors and compasses.

Less commonly found is the adjustable set square. Here, the body of the object is cut in half and rejoined with a hinge marked with angles. Adjustment to the marked angle will produce any desired angle up to a maximum of 180°.

Geodreieck

[edit]
Original Geodreieck
Variant of larger TZ-Dreieck with extra markings

In some European countries a common form of set square combines a 90-45-45 triangle, a ruler and a protractor into a single tool made of stiff or slightly flexible transparent plastic.[1][2][3] Being a mandatory tool used by pupils in middle school and higher in German-speaking and neighbouring countries, this specific design is named "Geodreieck" (short form of "Geometrie-Dreieck", meaning "geometry triangle") or similar.[3] It was originally developed in 1964 by the German manufacturer Dennert & Pape Aristo-Werke [de] (after several refirmations now Geotec Schul- und Bürowaren GmbH).[4][5][6][3][7] Relatively uncommon in English-speaking countries, this is sometimes called a "protractor triangle", a term, however, also used for other similar designs. The original design has a hypotenuse length of 15.8 cm and features a 2×7 cm symmetry scale in millimeter and degree raster.[3] Variants in larger sizes, with fixed or detachable handles, with or without bevelled edges (facets), and with or without ink nodules or embossed labels exist as well. Some variants have extra markings at angles of 7° and 42° (138° and 173°) in addition to the normal 45° and 90° markings to ease dimetric axonometry per ISO 5456-3,[8] others feature angle scales in gons instead of degrees. Several other somewhat similar designs named "TZ-Dreieck" ("TZ triangle") exist for (larger) technical drawings (TZ from German: Technisches Zeichnen). The scale reaches from 10 to 10 cm, or even 11 to 11 cm.

[edit]
Navigational Cras protractor (foreground)

Similar to the Geodreieck, a number of other protractor triangle types exist for navigation purposes. Various designs are named navigation (protractor) triangle, nautical navigational triangle, nautical set square, Portland (navigational) triangle or Portland protractor triangle, Kent-type triangle, Inoue-type A/B nautical triangle or plotting triangle, course triangle, yachtsmen triangle, and supporting triangle.

However, some widely used navigational protractors, such as the Cras protractor, are not triangles.


See also

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Set square

View on Grokipedia
from Grokipedia
A set square, also known as a in , is a fundamental drafting instrument used in , , and to create precise straight lines at right angles (90 degrees) or other fixed angles such as 30°, 45°, or 60° relative to a baseline, and to verify perpendicularity between lines. Typically shaped as an isosceles or right-angled , or an L-shaped form, it serves as a guide for pencils or pens when placed against a like a or , ensuring accuracy in geometric constructions without freehand sketching. Originating from ancient tools for measuring right angles—such as those implied in early practices—set squares evolved into more refined instruments by the , with hinged or folding designs appearing in by makers like Nicholas Bion (1652–1733) and the Chapotot family (late 17th to early ). Early examples, often crafted from wood or metal with etched scales for , were essential for fields like design, fortification planning (as seen in pre-1876 Japanese ho kaku-gi tools), and post-Revolutionary French metric adaptations around 1800. By the 19th and 20th centuries, set squares became standardized components of kits, transitioning to durable plastics for school and professional use, though traditional and wooden variants persist in specialized applications. In modern contexts, set squares facilitate the creation of vertical, horizontal, and diagonal lines, supporting tasks in mechanical drafting, , and layout, often paired with protractors for adjustable angles or plumb bobs for vertical alignment in . Their simplicity and portability have made them enduring tools, even as (CAD) software has supplemented manual methods since the late , preserving their role in and precision handwork.

Definition and purpose

Description

A set square is a triangular plate used in technical drawing for creating right angles, parallel lines, and specific angles. It is typically formed as a right-angled triangle with two legs of equal or unequal length and a hypotenuse; the edges are straight and often calibrated with measurement graduations for precision. The geometric properties include a 90° angle at one vertex, with the other angles at the vertices being 45° (in isosceles variants) or 30° and 60° (in other variants depending on the type); the hypotenuse is frequently employed to draw 45° or additional angles when aligned against a straightedge. Common dimensions along the legs measure 10 cm, 15 cm, or 25 cm.

Common uses

Set squares are primarily employed in technical drawing to produce perpendicular, parallel, and angled lines by aligning one edge with a T-square or straightedge, ensuring precision in layouts and diagrams. This alignment allows draftsmen to draw vertical lines directly from the T-square's working edge and inclined lines at standard angles such as 30°, 45°, or 60° by positioning the set square's hypotenuse or leg accordingly. In educational settings, set squares facilitate the teaching of geometry by enabling students to construct basic shapes like triangles, squares, and polygons, thereby enhancing visualization and performance in geometric tasks. For instance, they assist in drawing right angles and parallel lines, which are fundamental for hands-on learning in secondary school curricula. In and , set squares support the creation of plans, diagrams, and blueprints by providing accurate angular references for structural layouts and schematic designs. They are integral to drafting processes where precise perpendiculars and parallels are required to scale models or elevations. Beyond these fields, set squares find application in for marking right angles and cuts prior to joint fabrication, ensuring alignment in assemblies. In , they aid pattern drafting by squaring off corners and drawing angled seams for garments like pants and dresses. Additionally, in navigation, set squares assist with plotting by measuring bearings and directions on nautical maps. Common techniques include sliding the set square along a or to generate while maintaining consistent spacing. The can also serve as a guide for diagonal measurements, particularly in layouts requiring 45° inclinations from isosceles variants.

Types

Isosceles right triangle set square

The isosceles set square, commonly referred to as the 45°-45°-90° set square, features a based on a where the two legs are of equal , forming two 45° angles adjacent to the at the vertex. This geometric configuration ensures the serves as the longest edge, providing a stable base for alignment during use. In drafting applications, this set square is ideal for constructing 45° lines, perpendiculars, squares, and diagonals by sliding it along a or to guide the or . Its enables quick 45° measurements from either leg, offering an advantage in efficiency for repetitive angular tasks compared to adjustable tools. Additionally, it finds common use in pictorial drawings such as cabinet projections, where 45° receding lines simulate depth in 3D representations. In basic layouts, variants like the speed square—shaped as a 45°-45°-90° triangle with a lipped edge—facilitate marking right angles for framing, 45° miter cuts for trim, and diagonal checks for squaring foundations. These set squares typically come in sizes ranging from 10 cm to 25 cm along the legs or , with common options including 15 cm (6 inches) for portable use and larger 8-inch models for broader layouts. Markings often include metric or imperial scales etched along the edges for measuring lengths, alongside degree indicators at the 45° and 90° angles to aid precise alignment. Unlike the 30°-60°-90° set square, which supports asymmetric angles for diverse geometric constructions, the isosceles variant excels in symmetric applications like square-based diagrams.

30-60-90 set square

The 30-60-90 set square is a scalene characterized by measuring 30°, 60°, and 90°. Its sides follow a fixed of 1 : √3 : 2, where the side opposite the 30° serves as the reference length, the side opposite the 60° is √3 times that length, and the is twice the shortest side. In and engineering applications, this set square facilitates the of 30° and 60° when used alongside a for vertical and horizontal alignment. It is particularly valuable for drawing equilateral triangles by aligning the 60° repeatedly and for creating regular hexagons through sequential 60° increments, as well as for rendering precise angled lines in mechanical schematics and supporting trigonometry-based exercises. A primary advantage of the 30-60-90 set square lies in its provision of exact 30° and 60° acute angles, enabling geometric constructions beyond the symmetric 45° capabilities of isosceles variants. These tools are commonly produced in sizes from 4 to 12 inches to suit various drafting scales, often incorporating etched or printed measurement markings along the edges in both inches and centimeters, along with labeled indicators for the 30°, 60°, and 90° angles to aid in rapid identification and use. When combined with a 45-45-90 set square, it expands options for comprehensive drafting tasks.

Specialized variants

Geodreieck

The Geodreieck ( triangle) is a multifunctional set square originating from , particularly popular in German-speaking regions. It combines a 45°-45°-90° triangle with an integrated semi-circular protractor, dual ruler scales, and parallel line guides, allowing for measurements up to 180°, use, and constructions in a single tool. Commonly used in and for its versatility and compact design.

Navigation protractor triangles

Navigation protractor triangles are specialized tools for nautical and aeronautical charting, typically featuring a right-angled (often 45°-45°-90°) with protractor degree markings along the and legs for measuring and plotting bearings, courses, and angles on maps or charts. They enable precise direction calculations without additional instruments and are essential for safe practices.

Specialized variants

Geodreieck

The Geodreieck is a specialized geometric drawing tool originating from , designed as a multifunctional set square for educational and technical applications. It features a right-angled isosceles triangular shape with 45°-45°-90° angles, constructed from transparent plexiglas (PMMA) for visibility during use. The typically measures 15.5 cm to 16 cm, incorporating a centimeter scale divided into millimeters along one edge, while the interior includes protractor markings graduated in 1° increments from 0° to 90° on the legs and up to 180° along the for versatile measurement and construction. Developed in 1964 by the Austrian company Dennert & Pape (operating as Aristo-Werke) in , the Geodreieck was created to streamline instruction in schools by integrating multiple drafting functions into a single, durable instrument. Production has continued at the same facility using injection-molded plexiglas, ensuring precision and transparency, with the tool becoming a staple in German-speaking educational systems from levels onward. Its design adheres to standards for straightness and accuracy, facilitating reliable use in classroom settings where it replaced separate rulers and protractors for many tasks. In educational contexts, the Geodreieck supports advanced constructions, such as drawing regular polygons by setting specific angles like 72° for pentagons or 60° for hexagons via its protractor scale, enabling students to create precise multi-sided figures without additional tools. It also aids in technical education for drafting using built-in scales and raised parallel edges, as well as constructing oblique angles for diagrams in subjects like physics and . Variants may include additional markings at 7° and 42° (or complementary 138° and 173°) to assist with axonometric projections in line with ISO 5456-3, enhancing its utility for three-dimensional representations. Unique features distinguish the Geodreieck from basic set squares, including beveled edges on all sides to guide pens without smearing, raised nubs (Tuschenoppen) for ink-resistant contact points, and optional ergonomic grips for prolonged use. These elements, combined with color-coded angle indicators, promote accuracy and ease in handling, particularly for younger students, with specialized "My First Geodreieck" versions scaled down to 16 cm for early learners. Its composite functionality—merging , protractor, and alignment aids—builds on traditional set square principles while expanding capabilities for complex drawings. Navigation protractor triangles are specialized drafting tools consisting of a triangular set square integrated with a protractor scale, primarily designed for precise angular measurements in nautical and aeronautical navigation. These instruments facilitate the plotting of bearings and courses directly on charts, combining the straightedge functionality of a set square with graduated angular markings for efficiency in dynamic environments like ship bridges or aircraft cockpits. The design typically features an isosceles base with 45°-45°-90° angles, measuring approximately 7 to 11 inches along the legs and up to 13 inches on the , allowing for stable placement on standard nautical charts. The protractor scale, inscribed along the , spans 0° to 180° for measuring true or magnetic bearings, often with additional cursors or directional indicators to align with north references and parallel rules. Constructed from clear acrylic or , the tool ensures visibility through the material when overlaid on maps, and some variants include a for easier manipulation during extended use. In applications, navigation protractor triangles are employed to plot courses on nautical charts by drawing bearing lines from position fixes, enabling mariners to calculate headings and track deviations from intended routes. In aviation, they aid in measuring azimuths on sectional charts for flight planning and en-route adjustments, frequently paired with dividers to scale distances accurately against chart projections. This combination supports and position verification in both maritime and aerial contexts, where quick angular transfers are critical for safety. A key unique feature is the transparent construction, which permits direct overlay on charts without hiding underlying details, enhancing precision in low-light or confined plotting spaces; while the primary scale uses degrees for standard navigation, select models incorporate radian markings for advanced trigonometric computations in specialized scenarios. These tools differ from basic set squares by embedding protractor functionality specifically for directional plotting rather than general geometric construction. Historically, protractor triangles evolved from 17th-century tools adapted for maritime use, where early protractors dating back to the were vital for laying out angles to determine ship directions and positions on charts. By the , their triangular form became standardized for in navigation, building on foundational instruments that revolutionized positional accuracy at sea.

History

Origins

The origins of set squares trace back to ancient civilizations, where rudimentary tools for constructing right angles were vital for geometric and architectural tasks. In , builders employed knotted ropes configured as 3-4-5 triangles to mark perpendicular lines during the construction of pyramids and other structures, a technique that ensured precise alignments without advanced instruments. Similarly, Babylonian mathematicians utilized Pythagorean triples—such as those documented on clay tablets from around 1800 BCE—for land surveying and division, demonstrating an early understanding of right-angled triangles that paralleled the functional principles of later set squares. Greek geometers, building on these foundations, primarily relied on straightedges and compasses for theoretical constructions as described by . During the medieval and Renaissance periods, these ancient concepts were integrated into more sophisticated mathematical toolsets in both Islamic and European traditions. In the Islamic world, particularly from the 9th to 15th centuries, scholars advanced geometric design using compasses and straight rulers to create intricate patterns and solve construction problems, laying groundwork for angle-measuring devices that emphasized precision in architecture and astronomy. This knowledge transmitted to Europe via translations during the Renaissance, where drawing instruments evolved to include early forms of set squares alongside compasses and rulers, aiding artists, architects, and mathematicians in replicating right angles and other geometric figures. By the 17th century, set squares had emerged as standardized components of scientific instrument kits in Europe, particularly for surveying and navigation, where portability and accuracy were paramount. Instrument makers like Nicolas Bion in Paris produced sets that bundled set squares with other tools for fieldwork, allowing users to draw angles directly on charts and plans. A key milestone was the development of hinged triangular set squares in the late 16th century, which folded for compact storage in drafting kits, marking the transition from ad hoc ancient methods to purpose-built implements. These innovations in the 16th and 17th centuries established the triangular form that would evolve into modern variants. Early examples, often crafted from wood or metal with etched scales for measurement, were essential for fields like artillery design and fortification planning, as seen in pre-1876 Japanese ho kaku-gi tools.

Modern development

The industrialization of set square production in the 19th century marked a significant shift from handcrafted tools to mass manufacturing, primarily driven by advancements in Britain and to meet growing demands in and engineering. In Britain, William Ford Stanley established his firm in 1853, initiating large-scale production of precision drawing instruments, including set squares, which made them more accessible for technical drafting and school use. Similarly, in , Gebrüder Haff began manufacturing drawing instruments, such as compasses and set squares, from 1835 onward, leveraging industrial techniques to supply educational institutions and engineers across . Post-Revolutionary French adaptations around 1800 incorporated metric scales into set squares, aligning with the new measurement system for and . In the , innovations focused on material advancements and specialized designs to enhance durability and functionality. In the mid-20th century, set squares transitioned to lightweight, transparent acrylic , replacing heavier and metal variants for better visibility and portability in drafting. A notable development occurred in the with the creation of the Geodreieck, a multifunctional tool combining a set square, protractor, and , invented in 1964 by designers at ARISTO in , , and quickly adopted in German-speaking educational systems. Standardization efforts in the late ensured consistency in set square dimensions and angles for educational and professional applications. The (ISO) introduced ISO 9960-3 in 1994, specifying requirements for fixed 45° and 60° set squares, as well as adjustable models, to promote uniformity in drawing-office tools. In recent decades, while (CAD) software has emerged as a primary digital alternative to manual set squares since the , reducing reliance on physical tools in professional engineering, traditional set squares remain prevalent in educational settings and hands-on drafting for their tactile precision and low cost. This persistence is evident in specialized variants, such as those adapted for protractors, which continue to bridge analog and digital workflows.

Materials and construction

Materials

Set squares have traditionally been constructed from in their early forms, providing a lightweight and easily workable material for basic drafting needs. In professional applications, metals such as or have been favored for their superior durability and resistance to wear, ensuring precision over extended use in environments. Contemporary set squares predominantly utilize clear acrylic or , which offer transparency for accurate overlay work and reduced weight for portability. is occasionally employed as an alternative for its flexibility, particularly in adjustable variants. These modern materials are selected for their shatter-resistant qualities, non-warping stability under varying conditions, and optical clarity, enhancing usability in both educational and technical settings. Material choices vary by application, with metals like preferred for heavy-duty tasks requiring robustness, while lightweight plastics such as acrylic are common in kits for cost-effectiveness and ease of handling.

Manufacturing processes

The manufacturing of set squares begins with the design phase, where (CAD) software is employed to model precise angles, such as 30°-60°-90° or 45°-45°-90°, and incorporate measurement scales for accuracy in drafting applications. This digital modeling ensures dimensional consistency and allows for simulations to verify structural integrity before production. For mass-produced plastic set squares, injection molding is the primary method, involving the melting of materials like acrylic or and injecting them into precision molds to form the triangular shape with integrated edges. The process yields high-volume output with consistent geometry, suitable for educational and professional tools. Metal variants, often made from or aluminum, are typically produced through die-cutting or stamping from sheet stock to outline the , followed by or to add durable scales and markings. This method provides robust edges for heavy-duty use in . Custom or small-batch set squares, particularly in acrylic, may utilize to precisely shape the material from sheets, enabling intricate designs and while maintaining clean edges. involves rigorous of edges and angles, often to tolerances of ±0.5° or better, using optical comparators or coordinate measuring machines to ensure squareness and prevent deviations that could affect drawing precision. Edges are inspected for sharpness and relief, typically chamfered to depths of 0.003" to 0.010" to resist cracking and ink buildup. During assembly, scales are added through , etching, or die for permanence, followed by beveling select edges to facilitate flow and handling. Completed units are then packaged, often in sets with protective cases for educational distribution.

References

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