What is a 45-45-90 triangle?

A 45-45-90 triangle is an isosceles right triangle with two equal legs. Its side ratio is 1 : 1 : sqrt(2), so the hypotenuse is the leg times sqrt(2).

What is the 30-60-90 triangle ratio?

The sides are in the ratio 1 : sqrt(3) : 2 for the short leg, long leg and hypotenuse. For a short leg of 5, the long leg is 5*sqrt(3), about 8.6603, and the hypotenuse is 10.

How do you find the hypotenuse of a 45-45-90 triangle?

Multiply the leg by sqrt(2): c = a x sqrt(2). For a leg of 7, the hypotenuse is 7*sqrt(2), about 9.8995.

How do you find a leg of a 45-45-90 triangle from the hypotenuse?

Divide the hypotenuse by sqrt(2): a = c / sqrt(2). For c = 10, the leg is 10 / sqrt(2) = 5*sqrt(2), about 7.0711.

How do you find the sides of a 30-60-90 triangle from the hypotenuse?

The short leg is half the hypotenuse and the long leg is the short leg times sqrt(3). For a hypotenuse of 12, the short leg is 6 and the long leg is 6*sqrt(3), about 10.3923.

Why are these called special right triangles?

Their side ratios are exact, so you do not need a calculator or trigonometry. The 45-45-90 triangle is half of a square cut along its diagonal, and the 30-60-90 triangle is half of an equilateral triangle cut along its height.

Special Right Triangles: 45-45-90 & 30-60-90

Special Right Triangles Reference Chart (45-45-90 & 30-60-90)

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Special right triangles have fixed side ratios, so missing sides need no trigonometry. This chart lists the 45-45-90 ratio 1 : 1 : root 2 and the 30-60-90 ratio 1 : root 3 : 2, with a live calculator.

45-45-90 Ratio1 : 1 : √2Isosceles right triangle

30-60-90 Ratio1 : √3 : 2Short leg : long leg : hypotenuse

45-45-90 Hypotenusec = a × √2Needs: one leg a

45-45-90 Lega = c / √2Needs: the hypotenuse c

30-60-90 Hypotenusec = 2 × aNeeds: the short leg a

30-60-90 Long Legb = a × √3Needs: the short leg a

Special Right Triangle Side Calculator

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Length

Tip: open this page with ?triangle=30-60-90 or ?triangle=45-45-90 to start with that triangle. Sides are always shown as a : b : c, where c is the hypotenuse (for 30-60-90, a is the short leg and b the long leg).

Special Right Triangles Reference Chart
Triangle Type	Angles	Side Ratios (a : b : c)	Hypotenuse Rule	Leg Rule	Copy
45°-45°-90°	45°, 45°, 90°	1 : 1 : √2	Hypotenuse = Leg × √2	Leg = Hypotenuse / √2
30°-60°-90°	30°, 60°, 90°	1 : √3 : 2	Hypotenuse = 2 × Short Leg	Long Leg = Short Leg × √3

45°-45°-90° Triangle: Solved for Legs 1 to 20
Leg (a = b)	Hypotenuse = a × √2	Exact Hypotenuse (c)	Hypotenuse (decimal)	Area (a² / 2)
1	1 × √2	√2	1.4142	0.5
2	2 × √2	2√2	2.8284	2
3	3 × √2	3√2	4.2426	4.5
4	4 × √2	4√2	5.6569	8
5	5 × √2	5√2	7.0711	12.5
6	6 × √2	6√2	8.4853	18
7	7 × √2	7√2	9.8995	24.5
8	8 × √2	8√2	11.3137	32
9	9 × √2	9√2	12.7279	40.5
10	10 × √2	10√2	14.1421	50
11	11 × √2	11√2	15.5563	60.5
12	12 × √2	12√2	16.9706	72
13	13 × √2	13√2	18.3848	84.5
14	14 × √2	14√2	19.7990	98
15	15 × √2	15√2	21.2132	112.5
16	16 × √2	16√2	22.6274	128
17	17 × √2	17√2	24.0416	144.5
18	18 × √2	18√2	25.4558	162
19	19 × √2	19√2	26.8701	180.5
20	20 × √2	20√2	28.2843	200

30°-60°-90° Triangle: Solved for Short Legs 1 to 20
Short Leg (a)	Exact Long Leg (a√3)	Long Leg (decimal)	Hypotenuse (2a)	Area (a²√3 / 2)
1	√3	1.7321	2	1√3 / 2 = 0.8660
2	2√3	3.4641	4	2√3 = 3.4641
3	3√3	5.1962	6	9√3 / 2 = 7.7942
4	4√3	6.9282	8	8√3 = 13.8564
5	5√3	8.6603	10	25√3 / 2 = 21.6506
6	6√3	10.3923	12	18√3 = 31.1769
7	7√3	12.1244	14	49√3 / 2 = 42.4352
8	8√3	13.8564	16	32√3 = 55.4256
9	9√3	15.5885	18	81√3 / 2 = 70.1481
10	10√3	17.3205	20	50√3 = 86.6025
11	11√3	19.0526	22	121√3 / 2 = 104.7891
12	12√3	20.7846	24	72√3 = 124.7077
13	13√3	22.5167	26	169√3 / 2 = 146.3583
14	14√3	24.2487	28	98√3 = 169.7410
15	15√3	25.9808	30	225√3 / 2 = 194.8557
16	16√3	27.7128	32	128√3 = 221.7025
17	17√3	29.4449	34	289√3 / 2 = 250.2813
18	18√3	31.1769	36	162√3 = 280.5922
19	19√3	32.9090	38	361√3 / 2 = 312.6352
20	20√3	34.6410	40	200√3 = 346.4102

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Frequently Asked Questions

Question	Answer
What is a 45-45-90 triangle?	A 45-45-90 triangle is an isosceles right triangle with two equal legs. Its side ratio is 1 : 1 : sqrt(2), so the hypotenuse is the leg times sqrt(2).
What is the 30-60-90 triangle ratio?	The sides are in the ratio 1 : sqrt(3) : 2 for the short leg, long leg and hypotenuse. For a short leg of 5, the long leg is 5*sqrt(3), about 8.6603, and the hypotenuse is 10.
How do you find the hypotenuse of a 45-45-90 triangle?	Multiply the leg by sqrt(2): c = a x sqrt(2). For a leg of 7, the hypotenuse is 7*sqrt(2), about 9.8995.
How do you find a leg of a 45-45-90 triangle from the hypotenuse?	Divide the hypotenuse by sqrt(2): a = c / sqrt(2). For c = 10, the leg is 10 / sqrt(2) = 5*sqrt(2), about 7.0711.
How do you find the sides of a 30-60-90 triangle from the hypotenuse?	The short leg is half the hypotenuse and the long leg is the short leg times sqrt(3). For a hypotenuse of 12, the short leg is 6 and the long leg is 6*sqrt(3), about 10.3923.
Why are these called special right triangles?	Their side ratios are exact, so you do not need a calculator or trigonometry. The 45-45-90 triangle is half of a square cut along its diagonal, and the 30-60-90 triangle is half of an equilateral triangle cut along its height.

Special Right Triangles Reference Chart (45-45-90 & 30-60-90)

Special Right Triangle Side Calculator

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Frequently Asked Questions

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