{"id":37690,"date":"2024-11-20T07:45:11","date_gmt":"2024-11-19T23:45:11","guid":{"rendered":"https:\/\/www.wukongsch.com\/blog\/?p=37690"},"modified":"2026-07-17T17:59:53","modified_gmt":"2026-07-17T09:59:53","slug":"30-60-90-triangle","status":"publish","type":"post","link":"https:\/\/www.wukongsch.com\/blog\/30-60-90-triangle-post-37690\/","title":{"rendered":"30-60-90 Triangle: Rules, Formula, Calculator, Examples & SAT Tips"},"content":{"rendered":"
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A 30-60-90 triangle is a special right triangle with angles of 30\u00b0, 60\u00b0, and 90\u00b0. Its sides follow a fixed ratio of 1 : \u221a3 : 2<\/strong>. The shortest leg (opposite the 30\u00b0 angle) equals x<\/strong>, the longer leg (opposite the 60\u00b0 angle) equals x\u221a3<\/strong>, and the hypotenuse (opposite the 90\u00b0 angle) equals 2x<\/strong>. You can solve for any missing side if you know just one side length. This triangle appears frequently in geometry, trigonometry, and standardized tests like the SAT and ACT.<\/p>\n\n\n

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<\/span>What Is a 30-60-90 Triangle?<\/span><\/h2>\n\n\n\n

A 30-60-90 triangle is a right triangle with three specific angles: 30 degrees, 60 degrees, and 90 degrees. It belongs to the family of special right triangles because its side lengths follow a consistent, predictable pattern. You do not need the Pythagorean theorem every time \u2014 you can use the ratio instead.<\/p>\n\n\n

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The 30\u00b0 angle sits opposite the shortest side. The 60\u00b0 angle sits opposite the medium-length side. The 90\u00b0 angle sits opposite the longest side, called the hypotenuse. This angle-to-side relationship never changes, which makes the triangle easy to work with.<\/p>\n

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You can think of a 30-60-90 triangle as half of an equilateral triangle cut along its altitude. When you split an equilateral triangle (all sides equal, all angles 60\u00b0) down the middle, you get two congruent 30-60-90 triangles. This geometric relationship explains why the side ratio works the way it does.<\/p>\n\n\n\n

<\/span>\u26a1 Quick Reference: 30-60-90 Triangle at a Glance<\/span><\/h2>\n\n\n\n
Property<\/td>Value<\/td><\/tr>
Angles<\/td>30\u00b0 \/ 60\u00b0 \/ 90\u00b0<\/td><\/tr>
Side Ratio<\/td>1 : \u221a3 : 2<\/td><\/tr>
Short Leg (x)<\/td>Opposite 30\u00b0<\/td><\/tr>
Long Leg (x\u221a3)<\/td>Opposite 60\u00b0<\/td><\/tr>
Hypotenuse (2x)<\/td>Opposite 90\u00b0<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Memory trick:<\/strong> “Short = x, Long = x\u221a3, Hyp = 2x.”<\/p>\n\n\n\n

\"60<\/figure>\n\n\n\n

<\/span>\ud83e\uddee Interactive Side Calculator<\/span><\/h2>\n\n\n\n

This article provides an interactive tool that instantly calculates the missing sides of a 30-60-90 triangle. You can enter any one side length \u2013 short leg, long leg, or hypotenuse \u2013 and the tool will output the other two sides using the 1:\u221a3:2 ratio. It also computes area and perimeter automatically. This saves time on homework and reduces arithmetic errors with radicals.<\/p>\n\n\n\n\n \n \n

30-60-90 Triangle Calculator<\/h3>\n