Special right triangles are the two right triangles whose angles are 45°-45°-90° or 30°-60°-90°. Their sides always come in fixed ratios: x : x : x√2 for the 45-45-90 triangle and x : x√3 : 2x for the 30-60-90 triangle. From one side you can find the other two without the Pythagorean theorem.
In Spanish
triángulos rectángulos especiales
Also called
special right triangle, 45-45-90 and 30-60-90 triangles, special triangles
A 45-45-90 triangle is half of a square, so its legs match. A 30-60-90 triangle is half of an equilateral triangle, so its short leg is half the hypotenuse.
Formal definition and notation
A right triangle is special when its acute angles make its sideratios exact. In a 45°-45°-90° triangle the legs are equal and the hypotenuse is √2 times a leg. In a 30°-60°-90° triangle the hypotenuse is twice the shorter leg, and the longer leg is √3 times the shorter leg. By similarity, these ratios hold for every triangle with those angles.
Where they come from: cut a square along its diagonal to get two 45-45-90 triangles; cut an equilateral triangle down its middle to get two 30-60-90 triangles.
Find the short side first (x), then multiply to get the others. If you know a longer side, divide to get back to x.
Check: the ratios agree with the Pythagorean theorem, since x2 + x2 = (x√2)2 and x2 + (x√3)2 = (2x)2.
They give the exact trig values used everywhere: sin 30° = 12, cos 45° = √22, tan 60° = √3.
The SAT reference sheet lists both triangles, so test questions expect you to recognize them.
Side ratios of the special right triangles
Side ratios of the special right triangles
45-45-90: x : x : x√2; 30-60-90: x : x√3 : 2x
x
a leg of the 45-45-90 triangle, or the shorter leg (opposite 30°) of the 30-60-90 triangle
x√2
the hypotenuse of a 45-45-90 triangle, about 1.414x
x√3
the longer leg (opposite 60°) of a 30-60-90 triangle, about 1.732x
2x
the hypotenuse of a 30-60-90 triangle
The smallest side is always across from the smallest angle, and the hypotenuse is across from the right angle.
Special Right Triangles examples, step by step
Example 1: 45-45-90: from a leg
Warm-up
A 45-45-90 triangle has legs of 7 cm. How long is the hypotenuse?
In a 45-45-90 triangle, hypotenuse = leg × √2.
Hypotenuse = 7√2 cm.
Decimal: 7 × 1.414 ≈ 9.9 cm.
Answer: The hypotenuse is 7√2 ≈ 9.9 cm.
Example 2: 30-60-90: from the short leg
Core
In a 30-60-90 triangle, the side opposite 30° is 5. Find the other two sides.
The side opposite 30° is the short leg: x = 5.
Hypotenuse = 2x = 10.
Long leg = x√3 = 5√3 ≈ 8.66.
Check: 52 + (5√3)2 = 25 + 75, which is 100 = 102.
Answer: The hypotenuse is 10 and the long leg is 5√3 ≈ 8.66.
Example 3: Height of an equilateral triangle
Test-style
An equilateral triangle has sides of 12 inches. What is its height?
The height splits it into two 30-60-90 triangles with hypotenuse 12.
Short leg = 122 = 6 (half the base).
The height is the long leg: 6√3.
Decimal: 6 × 1.732 ≈ 10.39 inches.
Answer: The height is 6√3 ≈ 10.39 inches.
Common mistakes
✗ MistakeIn a 30-60-90 triangle with short leg 4, the hypotenuse is 4√3.
✓ InsteadThe hypotenuse is 2 × 4 = 8; the long leg is 4√3.
√3 ≈ 1.73 is less than 2, so x√3 is the middle side, not the longest. The hypotenuse is always 2x.
✗ MistakePutting x√3 opposite the 30° angle.
✓ InsteadOpposite 30° is the shortest side, x. Opposite 60° is x√3.
The smallest angle faces the smallest side, and the largest angle (90°) faces the largest side.
✗ MistakeAny right triangle with a 45° angle has sides 1, 1, √2, and so does one with sides 3, 4, 5.
✓ InsteadA 45° angle forces the other acute angle to be 45° too, but a 3-4-5 triangle has no 45° or 30° angle, so it is not special.
Only the two angle sets 45-45-90 and 30-60-90 give these ratios. Integer triples like 3-4-5 are a different idea.
The other special right triangle: sides x, x√3, 2x, with the hypotenuse exactly twice the short leg.
Special Right Trianglesvs.Pythagorean triple
A Pythagorean triple has whole-number sides, like 3, 4, 5. The special right triangles are defined by their angles, and their sides usually involve √2 or √3.
Quick check: 3 questions
Question 1 of 3
A 45-45-90 triangle has a hypotenuse of 10√2. How long is each leg?
Show answer and explanation
Answer: C. Hypotenuse = leg × √2, so leg = 10√2√2 = 10. 5√2 halves the hypotenuse as in a 30-60-90 triangle, and 20 multiplies instead of divides.
Question 2 of 3
In a 30-60-90 triangle the hypotenuse is 18. How long is the side opposite the 60° angle?
Show answer and explanation
Answer: B. Short leg x = 182 = 9, and the side opposite 60° is the long leg, 9√3. 9 is the side opposite 30°, and 18√3 is longer than the hypotenuse, which is impossible.
Question 3 of 3
Which set of side lengths belongs to a 30-60-90 triangle?
Show answer and explanation
Answer: D. With x = 3: x√3 = 3√3 and 2x = 6. The first set is a 45-45-90 triangle, 3-4-5 is not special, and in the third set the hypotenuse should be 4.
Special Right Triangles: frequently asked questions
Why are they called special right triangles?
Because their angles make the side ratios exact and easy to remember. Any right triangle can be solved with the Pythagorean theorem or trigonometry, but these two can be solved in your head from one side. They also give the exact sine, cosine and tangent of 30°, 45° and 60°, which trigonometry uses constantly.
How do I remember the 30-60-90 ratios?
Write the angles in order, 30, 60, 90, and the sides across from them in order: x, x√3, 2x. Think “1, 2, 3”: the numbers 1, 2 and √3 appear, and the biggest number, 2, goes with the biggest angle. Remember the picture too: half of an equilateral triangle, so the short leg is half the hypotenuse.
Can I just use the Pythagorean theorem instead?
Only if you already know two sides. The special ratios let you find two missing sides from just one side, because the angles give the shape. When you know two sides, the Pythagorean theorem is a good check: in a 30-60-90 triangle with short leg 5, 52 + (5√3)2 = 102.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 8.G.B.7, CCSS HSG-SRT.C.6, CCSS HSG-SRT.C.8
Cite this entry
Effortless Math Team. “Special Right Triangles.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/special-right-triangles/
Flashcard
TermSpecial Right Triangles45-45-90 and 30-60-90
DefinitionRight triangles with fixed side ratios: x, x, x√2 and x, x√3, 2x. Short leg 5 gives hypotenuse 10.
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Every triangle has at least two acute angles.Geometry · Grade 4, Grade 5, Grade 7SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2RatioA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 35. Order matters, and a ratio can compare part to part or part to whole. Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7, Grade 8TriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. 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One inch is exactly 2.54 centimeters, and 12 inches make 1 foot.Measurement · Grade 2, Grade 3, Grade 4HeightHeight is how tall something is: the distance from its bottom to its top, measured straight up and down. In geometry, the height of a shape is the perpendicular distance from a base to the opposite vertex or side, so it always meets the base at a right angle. Area and volume formulas use it.Measurement · Grade 2, Grade 5, Grade 6TrigonometryTrigonometry is the branch of math that studies how the angles and the side lengths of triangles are related. Its three main ratios, sine, cosine and tangent, compare two sides of a right triangle. Knowing one side and one acute angle, or two sides, you can find every other side and angle without measuring them.Trigonometry · Geometry, Algebra 2, PrecalculusSquare RootA square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.Number System · Grade 8, Algebra 1SineSine is the trigonometric ratio that compares the side opposite an angle with the hypotenuse of a right triangle: sin θ = oppositehypotenuse. On the unit circle, sin θ is the y-coordinate of the point at angle θ, so sine values always stay between −1 and 1.Trigonometry · Geometry, Algebra 2, PrecalculusUnit CircleThe unit circle is the circle with radius 1 centered at the origin, x2 + y2 = 1. An angle θ measured from the positive x-axis meets the circle at the point (cos θ, sin θ). This lets sine and cosine work for every angle, including obtuse and negative angles, not just the acute angles of right triangles.Trigonometry · Algebra 2, PrecalculusSimilar FiguresSimilar figures have the same shape but not necessarily the same size. Corresponding angles match, and corresponding side lengths are proportional: each side of one figure is the same multiple, the scale factor, of the matching side of the other. A scale drawing is similar to the real object.Geometry · Grade 7, Grade 8, Algebra 1Triangle Inequality TheoremThe triangle inequality theorem says that in any triangle, the sum of the lengths of any two sides is greater than the length of the third side. So 3, 4 and 5 can form a triangle, but 2, 3 and 6 cannot, because 2 + 3 < 6. Shortcut: the two shorter sides must add to more than the longest.Geometry · Grade 7, Geometry