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30 60 90 Triangle: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">30 60 90 Triangle30 60 90 Triangle: Definition and ExamplesTable of Contents30-60-90 Triangle Definition of 30-60-90 Triangle

A 30−60−9030-60-9030−60−90 triangle is a special right triangle with angles measuring 30°30°30°, 60°60°60°, and 90°90°90°. The angles of this triangle are in the ratio 1:2:31:2:31:2:3. In this special triangle, the side opposite to the 30°30°30° angle is the shortest (also called the shortest leg), the side opposite to the 60°60°60° angle is the longer leg, and the side opposite to the 90°90°90° angle is the largest side, known as the hypotenuse.

The sides of a 30−60−9030-60-9030−60−90 triangle follow a constant relationship and are always in the ratio of 1:3:21:\sqrt{3}:21:3​:2. This means if we call the side opposite to the 30°30°30° angle as "a", then the side opposite to the 60°60°60° angle will be "a3\sqrt{3}3​", and the hypotenuse (side opposite to the 90°90°90° angle) will be "222a". This special relationship allows us to find any side of the triangle when we know just one side.

Examples of 30-60-90 Triangle Example 1: Finding a Side Length Using the Shortest Side Problem:

Find the length of the side BC in a 30−60−9030-60-9030−60−90 triangle where AB = 666 cm.

30 60 90 degree angle

Step-by-step solution:

Step 1, Look at what we know. We have the side opposite to the 30°30°30° angle (shortest side), AB = 666 cm.

Step 2, Use the 30−60−9030-60-9030−60−90 triangle side ratio. The sides of a 30−60−9030-60-9030−60−90 triangle are always in the ratio 1:3:21:\sqrt{3}:21:3​:2.

Step 3, Set up the side lengths using our known value. If AB = 666 cm (which is our "aaa" value), then:

BC (side opposite to 60°60°60°) = a3a\sqrt{3}a3​ = 636\sqrt{3}63​ cm AC (hypotenuse) = 2a2a2a = 121212 cm

Step 4, Write the answer. The length of side BC = 636\sqrt{3}63​ cm.

Example 2: Finding the Hypotenuse Using the Middle Side Problem:

Find the length of the hypotenuse in a 30−60−9030-60-9030−60−90 triangle where QR = 838\sqrt{3}83​ cm.

30 60 90 degree angle

Step-by-step solution:

Step 1, Look at what we know. We have the side opposite to the 60°60°60° angle, QR = 838\sqrt{3}83​ cm.

Step 2, Understand which formula to use. When the side opposite to 60°60°60° (middle side) is given, the hypotenuse equals 2a3\frac{2a}{\sqrt{3}}3​2a​ where "aaa" is the given side.

Step 3, Substitute our known value. QR = aaa = 838\sqrt{3}83​ cm, so the hypotenuse:

PR = 2a3=2×833\frac{2a}{\sqrt{3}} = \frac{2 \times 8\sqrt{3}}{\sqrt{3}}3​2a​=3​2×83​​

Step 4, Simplify the expression:

PR = 1633=16\frac{16\sqrt{3}}{\sqrt{3}} = 163​163​​=16 cm

Step 5, Write the answer. The length of the hypotenuse PR = 16 cm.

Example 3: Verifying a Triangle is a 30-60-90 Triangle Problem:

A triangle has sides 323\sqrt{2}32​, 363\sqrt{6}36​, and 383\sqrt{8}38​. Find the angles of this triangle.

triangle

Step-by-step solution:

Step 1, Check if the sides match the 30−60−9030-60-9030−60−90 triangle ratio (1:3:21:\sqrt{3}:21:3​:2). To do this, divide each side by the smallest side.

Step 2, Find the smallest side. The sides are 323\sqrt{2}32​, 363\sqrt{6}36​, and 383\sqrt{8}38​.

32=3×1.414...=4.24...3\sqrt{2} = 3 × 1.414... = 4.24...32​=3×1.414...=4.24...

36=3×2.449...=7.35...3\sqrt{6} = 3 × 2.449... = 7.35...36​=3×2.449...=7.35...

38=3×2.828...=8.48...3\sqrt{8} = 3 × 2.828... = 8.48...38​=3×2.828...=8.48...

So, 323\sqrt{2}32​ is the smallest side.

Step 3, Divide all sides by 323\sqrt{2}32​:

32÷32=13\sqrt{2} ÷ 3\sqrt{2} = 132​÷32​=1

36÷32=36÷2=33\sqrt{6} ÷ 3\sqrt{2} = 3\sqrt{6} ÷ \sqrt{2} = \sqrt{3}36​÷32​=36​÷2​=3​

38÷32=8÷2=23\sqrt{8} ÷ 3\sqrt{2} = \sqrt{8} ÷ \sqrt{2} = 238​÷32​=8​÷2​=2

Step 4, Compare the result with the 30−60−9030-60-9030−60−90 triangle ratio. We have 1:3:21:\sqrt{3}:21:3​:2, which matches!

Step 5, Write the answer. Since the sides follow the 30−60−9030-60-9030−60−90 triangle rule, the angles of the triangle are 30°30°30°, 60°60°60°, and 90°90°90°.

Comments(2)BBaseballFanaticScarlettNovember 4, 2025This clear def of 30 60 90 triangle really helped my students grasp the concept. Thanks for the useful resource!

NNatureLover85September 17, 2025I’ve been helping my kid with geometry, and this page made 30-60-90 triangles so easy to understand! The ratio trick is genius, and the examples were super helpful for homework practice.

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