“45 67.5 67.5 Triangle Clear Background” has recently appeared in design searches, educational queries, and geometry resource pages. It refers to a scalene triangle with angles measuring 45°, 67.5°, and 67.5°, often requested as a transparent PNG, vector, or clear-background diagram for math lessons, digital design, UI/UX mockups, or 3D modeling.
This article explains the geometry behind the triangle, why its angles matter, and how designers, teachers, and students use this shape.
What Is a 45-67.5-67.5 Triangle?
A 45-67.5-67.5 triangle is a scalene acute triangle, meaning all three angles are less than 90° and no two sides are equal. Its angle structure is:
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45°
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67.5°
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67.5°
This triangle is especially interesting because:
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It links to geometric halving, since 67.5° = 135° ÷ 2.
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Its angles appear frequently in polygon division, trigonometry, and rotational symmetry.
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It is sometimes used to explain non-standard triangles in math education.
When rendered as a clear-background image, the triangle becomes a useful graphic for presentation slides, worksheets, or design overlays without visible borders interfering with visual layout.
Why People Search for a “Clear Background” Version
A clear background (transparent PNG) or SVG version of this triangle is valuable for digital use because it can be placed on:
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Math worksheets
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Illustrations and diagrams
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Presentations
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Geometry tutorials
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User interface prototypes
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CAD or 3D modeling reference images
A transparent background allows the triangle to sit cleanly on any color or image without a white box around it. Designers often need 45-67.5-67.5 triangle PNGs for easy layering in programs like:
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Photoshop
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Illustrator
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Canva
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PowerPoint
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Google Slides
This makes the shape versatile for both educational and professional settings.
Geometric Properties of the 45-67.5-67.5 Triangle
The 45-67.5-67.5 triangle has several notable geometric properties:
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It is scalene, meaning every side is a different length.
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It is acute, because all angles are under 90°.
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The 67.5° angle is derived from a half-square (135° / 2).
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It can be constructed using bisected right-angle geometry.
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The ratio between sides can be calculated using the Law of Sines:
asin(45°)=bsin(67.5°)=csin(67.5°)\frac{a}{\sin(45°)} = \frac{b}{\sin(67.5°)} = \frac{c}{\sin(67.5°)}
In standard form, the side opposite the 45° angle is the longest.
This makes the triangle useful in trigonometry lessons or geometric constructions where fractional angles such as 22.5° and 67.5° are needed.
Uses in Math Education and Lesson Diagrams
Teachers often search for clear-background triangles to include in:
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Angle identification worksheets
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Triangle classification exercises
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Trigonometry lessons on sine, cosine, tangent
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Construction activities using compasses and protractors
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Supplementary/Complementary angle lessons
The 45-67.5-67.5 triangle is especially helpful because it:
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Reinforces the idea that triangles can have unusual angle combinations.
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Encourages students to explore non-right triangles.
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Demonstrates how angle bisectors can create new geometric shapes.
Many digital classrooms now rely on transparent PNG geometry diagrams for clean, professional-looking instructional materials.
Graphic Design & Digital Illustration Applications
Beyond mathematics, this specific triangle is used in design, modeling, and animation, especially when creators need:
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unique triangle motifs
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angled shapes for logos or icons
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reference geometry for 3D mesh construction
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polygon cuts for low-poly illustrations
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rotational symmetry guides for pattern creation
The 45-67.5-67.5 triangle is compatible with octagonal and 16-sided polygon divisions, which makes it useful for:
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architectural patterns
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game design
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mandala or radial symmetry design
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laser cutting templates
Because the triangle is uncommon, it provides a distinctive look compared to standard 30-60-90 or 45-45-90 triangles.
Where to Download or Create a 45-67.5-67.5 Triangle Clear Background
If you’re looking for a transparent version of this triangle, here are reliable options:
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Create one yourself using a free vector program (Inkscape, Figma).
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Use geometry generators or diagram websites that export PNG/SVG.
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Download from graphics repositories offering free math shapes.
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Generate using PowerPoint or Google Drawings by setting “transparent background” on export.
When generating your own, be sure to:
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Set the canvas background to transparent,
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Use precise angle tools,
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Export as PNG for transparency or SVG for scalability.
