A 45-45-90 triangle is a right triangle with two 45° angles and two equal legs. Its sides are always in the ratio 1 : 1 : √2: the hypotenuse is a leg times √2. Legs of 5 give a hypotenuse of 5√2. It is half of a square cut along a diagonal.
In Spanish
triángulo 45-45-90
Also called
isosceles right triangle, 45 45 90 triangle, right isosceles triangle, half-square triangle
Equal legs of 5 give a hypotenuse of 5√2. Going the other way, divide by√2.
Formal definition and notation
A 45°-45°-90° triangle is a right triangle whose acute angles are both 45°; equivalently, an isosceles right triangle. If each leg has lengtha, the Pythagorean theorem gives a hypotenuse c with c2 = a2 + a2 = 2a2, so c = a√2 and a = c√2 = c√22.
Symbol
Read it as
Meaning
a : a : a√2
a to a to a root 2
leg : leg : hypotenuse
c√22
c root 2 over 2
a leg found from hypotenuse c, with the radical moved out of the denominator
Key facts
Two equal legs, two equal angles. It is the only right triangle that is also isosceles.
Leg to hypotenuse: multiply by √2. Hypotenuse to leg: divide by √2, which is the same as multiplying by √22 ≈ 0.707.
Area = 12a2, since the legs are the base and the height.
Every square‘s diagonal makes two of these triangles: a 10-inch square has a diagonal of 10√2 ≈ 14.14inches.
It gives sin 45° = cos 45° = √22 and tan 45° = 1.
45-45-90 triangle formulas
45-45-90 triangle formulas
c = a√2 and a = c√2 = c√22
a
each leg (the two legs are equal)
c
the hypotenuse, across from the right angle
√2
about 1.414, so the hypotenuse is about 41% longer than a leg
The diagonal of a square with side s is s√2, because the diagonal splits the square into two 45-45-90 triangles.
Try it: keep the legs equal
Interactive
Drag the blue or orange point, or select one and use the arrow keys. While the two legs are equal, the hypotenuse is always a leg times the square root of 2. Make the legs unequal and the triangle is no longer 45-45-90.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
45-45-90 Triangle examples, step by step
Example 1: Hypotenuse from a leg
Warm-up
A 45-45-90 triangle has legs of 6 m. Find the hypotenuse.
Hypotenuse = leg × √2.
c = 6√2 m.
Decimal: 6 × 1.414 ≈ 8.49 m.
Answer: The hypotenuse is 6√2 ≈ 8.49 m.
Example 2: Leg from the hypotenuse
Core
The hypotenuse of a 45-45-90 triangle is 10. How long is each leg?
Leg = hyp√2 = 10√2.
Multiply top and bottom by √2: 10√22 = 5√2.
Decimal: 5 × 1.414 ≈ 7.07.
Check: each leg squared is (5√2)2 = 50, and 50 + 50 = 100 = 102.
Answer: Each leg is 5√2 ≈ 7.07.
Example 3: Square from its diagonal
Test-style
A square’s diagonal is 12 cm. What are its side length and area?
The diagonal cuts the square into two 45-45-90 triangles, with the diagonal as hypotenuse.
Side = 12√2 = 6√2 cm.
Area = (6√2)2 = 36 × 2 = 72 square cm.
Shortcut: area of a square = d22 = 1442 = 72.
Answer: The side is 6√2 ≈ 8.49 cm and the area is 72 square cm.
Common mistakes
✗ MistakeHypotenuse 10, so each leg is 10√2.
✓ InsteadEach leg is 10√2 = 5√2 ≈ 7.07.
The hypotenuse is the longest side, so a leg must be shorter. Multiply by √2 to go up to the hypotenuse; divide to come down.
✗ MistakeHypotenuse 10, so each leg is 10 ÷ 2 = 5.
✓ InsteadEach leg is 5√2, not 5.
Halving the hypotenuse is the 30-60-90 rule for the short leg. Check: 52 + 52 = 50, not 100.
✗ MistakeLeaving 10√2 and calling it different from 5√2.
✓ Instead10√2 = 10√22, which is 5√2: the same number.
Rationalizing the denominator rewrites the value without changing it. Many answer choices show only the simplified form.
In a 30-60-90 triangle the hypotenuse is twice the short leg. In a 45-45-90 triangle it is √2 times a leg, and the legs match.
45-45-90 Trianglevs.Isosceles triangle
Every 45-45-90 triangle is isosceles, but most isosceles triangles, such as one with angles 70°, 70° and 40°, have no right angle.
Quick check: 3 questions
Question 1 of 3
A 45-45-90 triangle has legs of 9. What is the hypotenuse?
Show answer and explanation
Answer: A. Hypotenuse = 9√2 ≈ 12.7. 18 uses the 30-60-90 doubling rule, 9√3 mixes in the other triangle, and 9√22 is shorter than a leg.
Question 2 of 3
The hypotenuse of a 45-45-90 triangle is 8√2. What is its area?
Show answer and explanation
Answer: C. Each leg is 8√2√2 = 8, so the area is 12 × 8 × 8 = 32. 64 forgets the one half, and 128 squares the hypotenuse.
Question 3 of 3
A square has sides of 15 feet. About how long is its diagonal?
Show answer and explanation
Answer: A. Diagonal = 15√2 ≈ 21.2 feet. 30 doubles the side, 26.0 uses √3, and 10.6 divides by √2 instead of multiplying.
45-45-90 Triangle: frequently asked questions
What is the ratio of sides in a 45-45-90 triangle?
1 : 1 : √2. The two legs are equal, and the hypotenuse is √2, about 1.414, times a leg. So legs of 3 give a hypotenuse of 3√2, and legs of 1 give √2. This follows from the Pythagorean theorem: a2 + a2 = 2a2.
Is a 45-45-90 triangle isosceles?
Yes. Its two base angles are both 45°, and equal angles sit across from equal sides, so the two legs have the same length. That makes it an isosceles right triangle, the only kind of right triangle that is isosceles. The equal sides are the legs, never the hypotenuse.
How do you find the legs if you know the hypotenuse?
Divide the hypotenuse by √2. For a hypotenuse of 14, each leg is 14√2 = 7√2 ≈ 9.9. A quick way: multiply the hypotenuse by √22, about 0.707. Then check that the legs are shorter than the hypotenuse. Both legs are always equal.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 8.G.B.7, CCSS HSG-SRT.C.8
Cite this entry
Effortless Math Team. “45-45-90 Triangle.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/45-45-90-triangle/
Flashcard
Term45-45-90 Triangle1 : 1 : √2
DefinitionAn isosceles right triangle: two equal legs and two 45° angles. Hypotenuse = leg × √2; legs of 5 give 5√2.
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