what is the formula to find the hypotenuse

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All right triangles accept one right (ninety-degree) angle, and the hypotenuse is the side that is opposite or the correct angle, or the longest side of the right triangle.^[1] The hypotenuse is the longest side of the triangle, and information technology's also very easy to find using a couple of different methods. This article will teach you how to find the length of the hypotenuse using the Pythagorean theorem when you know the length of the other two sides of the triangle. It volition then teach you lot to recognize the hypotenuse of some special correct triangles that often announced on tests. It volition finally teach yous to find the length of the hypotenuse using the Law of Sines when you lot only know the length of one side and the mensurate of one additional bending.

1. 1

   Acquire the Pythagorean Theorem. The Pythagorean Theorem describes the relationship between the sides of a right triangle.^[2] It states that for any right triangle with sides of length a and b, and hypotenuse of length c, a² + b² = c² .^[three]

2. 2

   Make sure that your triangle is a right triangle. The Pythagorean Theorem merely works on right triangles, and by definition only right triangles can have a hypotenuse. If your triangle contains i bending that is exactly 90 degrees, it is a correct triangle and you can proceed.

   • Right angles are often notated in textbooks and on tests with a small foursquare in the corner of the bending. This special marking means "90 degrees."

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3. iii

   Assign variables a, b, and c to the sides of your triangle. The variable "c" volition ever exist assigned to the hypotenuse, or longest side. Choose 1 of the other sides to be a, and telephone call the other side b (it doesn't affair which is which; the math will plough out the same). Then copy the lengths of a and b into the formula, according to the following instance:

   • If your triangle has sides of iii and 4, and you have assigned letters to those sides such that a = three and b = four, then you should write your equation out as: 3² + 4² = c² .

4. 4

   Find the squares of a and b. To find the square of a number, yous but multiply the number by itself, so a² = a 10 a. Find the squares of both a and b, and write them into your formula.

   • If a = 3, a^two = iii x 3, or 9. If b = 4, then bⁱⁱ = iv x iv, or xvi.
   • When you lot plug those values into your equation, it should at present look like this: 9 + 16 = c^two .

5. v

   Add together together the values of a² and b² . Enter this into your equation, and this will give you the value for cⁱⁱ. There is simply one footstep left to become, and you will have that hypotenuse solved!

   • In our instance, nine + 16 = 25, so you should write downwardly 25 = c² .

6. 6

   Find the square root of cⁱⁱ. Use the square root function on your calculator (or your memory of the multiplication table) to detect the foursquare root of c². The answer is the length of your hypotenuse!

   • In our example, c² = 25. The square root of 25 is 5 (5 10 5 = 25, and then Sqrt(25) = 5). That means c = 5, the length of our hypotenuse!

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1. one

   Larn to recognize Pythagorean Triple Triangles. The side lengths of a Pythagorean triple are integers that fit the Pythagorean Theorem. These special triangles appear oftentimes in geometry text books and on standardized tests similar the SAT and the GRE. If yous memorize the first 2 Pythagorean triples, in particular, y'all can salve yourself a lot of time on these tests because you can immediately know the hypotenuse of one of these triangles simply by looking at the side lengths! ^[4]

   • The commencement Pythagorean triple is 3-4-5 (3² + 4² = 5^two, 9 + 16 = 25). When you lot see a right triangle with legs of length iii and four, you tin can instantly be certain that the hypotenuse volition exist v without having to exercise whatsoever calculations.
   • The ratio of a Pythagorean triple holds truthful fifty-fifty when the sides are multiplied past another number. For example a right triangle with legs of length 6 and 8 will accept a hypotenuse of 10 (half-dozen² + 8² = xⁱⁱ, 36 + 64 = 100). The same holds true for nine-12-15, and even one.5-2-2.5. Try the math and see for yourself!
   • The 2nd Pythagorean triple that commonly appears on tests is 5-12-thirteen (5ⁱⁱ + 12² = thirteen², 25 + 144 = 169). Also be on the lookout for multiples similar 10-24-26 and 2.5-6-6.v.

2. 2

   Memorize the side ratios of a 45-45-xc right triangle. A 45-45-90 right triangle has angles of 45, 45, and 90 degrees, and is also called an Isosceles Correct Triangle. It occurs oftentimes on standardized tests, and is a very piece of cake triangle to solve. The ratio between the sides of this triangle is ane:one:Sqrt(2), which ways that the length of the legs are equal, and the length of the hypotenuse is merely the leg length multiplied by the square root of two.

   • To calculate the hypotenuse of this triangle based on the length of ane of the legs, simply multiply the leg length past Sqrt(two).
   • Knowing this ratio comes in especially handy when your test or homework question gives you the side lengths in terms of variables instead of integers.

3. three

   Learn the side ratios of a 30-60-90 right triangle. This triangle has angle measurements of 30, lx, and 90 degrees, and occurs when you cut an equilateral triangle in one-half. The sides of the 30-sixty-xc right triangle ever maintain the ratio ane:Sqrt(3):2, or 10:Sqrt(iii)x:2x. If you lot are given the length of 1 leg of thirty-sixty-90 right triangle and are asked to find the hypotenuse, it is very easy to do:^[5]

   • If y'all are given the length of the shortest leg (contrary the xxx-degree angle,) simply multiply the leg length by 2 to detect the length of the hypotenuse. For instance, if the length of the shortest leg is 4, you know that the hypotenuse length must be eight.
   • If you are given the length of the longer leg (opposite the sixty-degree angle,) multiply that length by ii/Sqrt(3) to observe the length of the hypotenuse. For instance, if the length of the longer leg is iv, y'all know that the hypotenuse length must be 4.62.

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1. i

   Empathize what "Sine" means. The terms "sine," "cosine," and "tangent" all refer to various ratios between the angles and/or sides of a right triangle. In a right triangle, the sine of an angle is defined equally the length of the side opposite the angle divided by the hypotenuse of the triangle. The abridgement for sine establish in equations and on calculators is sin .^[6]

2. two

   Acquire to summate sine. Even a basic scientific calculator will have a sine part. Look for a key marked sin . To find the sine of angle, you volition usually press the sin key then enter the angle measurement in degrees. On some calculators, still, you must enter the degree measurement first so the sin key. You will have to experiment with your figurer or cheque the manual to detect out which information technology is.

   • To find the sine of an 80 caste angle, you will either demand to key in sin fourscore followed by the equal sign or enter fundamental, or 80 sin . (The respond is -0.9939.)
   • You tin also type in "sine estimator" into a web search, and find a number of easy-to-use calculators that will remove any guesswork.^[7]

3. 3

   Learn the Law of Sines. The Law of Sines is a useful tool for solving triangles. In item, it tin can help you find the hypotenuse of a right triangle if you know the length of one side, and the measure of one other angle in add-on to the right angle. For any triangle with sides a, b, and c, and angles A, B, and C, the Police force of Sines states that a / sin A = b / sin B = c / sin C.^[viii]

   • The Constabulary of Sines can actually exist used to solve any triangle, just merely a right triangle will have a hypotenuse.

4. 4

   Assign the variables a, b, and c to the sides of your triangle. The hypotenuse (longest side) must be "c". For the sake of simplicity, characterization the side with the known length as "a," and the other "b". Then assign variables A, B, and C to the angles of the triangle. The right angle opposite the hypotenuse will exist "C". The angle reverse side "a" is angle "A," and the bending opposite side "b" is "B".

5. five

   Summate the measurement of the third angle. Because it is a right angle, you already know that C = 90 degrees, and you also know the measure of A or B. Since the internal degree measurement of a triangle must ever equal 180 degrees, you tin can hands summate the measurement of the third angle using the following formula: 180 – (ninety + A) = B. You tin as well reverse the equation such that 180 – (90 + B) = A.

   • For example, if y'all know that A = forty degrees, then B = 180 – (90 + xl). Simplify this to B = 180 – 130, and you lot tin can rapidly determine that B = fifty degrees.

6. 6

   Examine your triangle. At this point, you should know the degree measurements of all three angles, and the length of side a. It is now time to plug this information into the Constabulary of Sines equation to determine the lengths of the other two sides.

   • To continue our example, let'due south say that the length of side a = 10. Angle C = ninety degrees, angle A = forty degrees, and angle B = fifty degrees.

7. 7

   Apply the Police of Sines to your triangle. Nosotros just need to plug our numbers in and solve the following equation to decide the length of hypotenuse c: length of side a / sin A = length of side c / sin C. This might still look a bit intimidating, but the sine of 90 degrees is a abiding, and always equals 1! Our equation can thus be simplified to: a / sin A = c / 1, or but a / sin A = c.

8. 8

   Carve up the length of side a by the sine of angle A to find the length of the hypotenuse! Y'all can practise this in two separate steps, by first calculating sin A and writing it down, and so dividing by a. Or you lot tin can cardinal it all into the calculator at the same time. If y'all practise, remember to include parentheses afterward the division sign. For instance, key in either 10 / (sin forty) or x / (twoscore sin), depending on your calculator.

   • Using our example, we discover that sin twoscore = 0.64278761. To find the value of c, we merely split up the length of a by this number, and learn that 10 / 0.64278761 = 15.half dozen, the length of our hypotenuse!

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Add New Question

• Question

  How do you detect the length of the hypotenuse in the Pythagorean Theorem?

  

  Grace Imson, MA
  Math Instructor, City College of San Francisco

  Grace Imson is a math teacher with over 40 years of didactics experience. Grace is currently a math instructor at the City Higher of San Francisco and was previously in the Math Section at Saint Louis University. She has taught math at the elementary, eye, high schoolhouse, and college levels. She has an MA in Education, specializing in Assistants and Supervision from Saint Louis University.

  

  Math Instructor, Urban center Higher of San Francisco

  Good Respond

  

  Support wikiHow by unlocking this expert answer.

  One mutual mistake is forgetting to foursquare the terms. In the Pythagorean Theorem, all iii terms are squared. Many people go too fast and forget to find the square before the sum of 'a' and 'b,' which gives them an incorrect answer.

• Question

  Is there a calculator for finding the length of the hypotenuse?

  

  This answer was written by i of our trained team of researchers who validated it for accuracy and comprehensiveness.

  

  wikiHow Staff Editor

  Staff Answer

  

  Back up wikiHow by unlocking this staff-researched answer.

  Google provides a right angle triangle calculator that allows you to solve for the hypotenuse. Simply search for "hypotenuse calculator" and plug your numbers into the calculator at the summit of the search results. You lot can as well utilize the hypotenuse calculator at Omincalculator.com.

• Question

  How tin you lot find the length of the hypotenuse given the length of ane side and an angle?

  

  This respond was written by one of our trained team of researchers who validated it for accuracy and comprehensiveness.

  

  wikiHow Staff Editor

  Staff Answer

  

  Support wikiHow by unlocking this staff-researched reply.

  If you lot know you are dealing with a right triangle, so you already know that i of the angles is 90°. Since the angles must add together up to 180°, yous tin solve for the missing angle using the formula 90 + X = 180. Once y'all accept all three angles, y'all can use that information and the known length of one side to apply the police force of sines and notice the length of the hypotenuse.

• Question

  How can I notice the length of the hypotenuse if only all angles are given?

  

  Stephen Han

  Community Answer

  You cannot discover the length of the hypotenuse. You have to know at to the lowest degree one of the sides of the triangle.

• Question

  Can I discover the hypotenuse if I but take the area of the triangle?

  

  No, you would as well demand to know the length of at to the lowest degree ane of the other sides.

• Question

  How do I find the expanse of a triangle if only the length of the hypotenuse is given?

  

  To get the area of a triangle, you need the base and the height or the length of all three sides. Since the angles are unknown, it is impossible.

• Question

  How do I observe the hypotenuse when I'm just given one side and surface area?

  

  Multiply the area by 2. Divide that by the known side. That gives yous the other not-hypotenuse side. Detect the square of each known side. Find the sum of those squares. Find the square root of that sum. That'southward the hypotenuse.

• Question

  How practice I find the iii angles of a triangle when but the three sides are given?

  

  Y'all would utilize the trigonometric functions (sine, cosine and tangent).

• Question

  Tin can I detect the hypotenuse if I only take the length of the two other sides?

  

  Yeah. Utilize the Pythagorean theorem.

• Question

  Can I utilise slope to find the hypotenuse?

  

  Yes, if yous're prepared to employ trigonometry to the question.

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Article Summary X

If you need to find the length of the hypotenuse of a correct triangle, you can apply the Pythagorean theorem if yous know the length of the other two sides. Foursquare the length of the 2 sides, called a and b, then add them together. Take the square root of the result to become the hypotenuse. If y'all want to larn how to find the hypotenuse using trigonometric functions, go along reading the article!

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