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Pythagorean Theorem Worksheets

This page has printables for teaching students about the Pythagorean Theorem. On these sheets, students will be given right angles. They'll find the lengths of unknown sides, or the hypotenuse.

Pythagorean Theorem Worksheets

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This page will link you to worksheets on algebra, integers, inequalities, and more.

Calculate the areas of rectangles, triangles, trapezoids, circles, and parallelograms.

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Pythagorean Theorem - Find Missing Side - Whole Numbers

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Find the missing side of a right triangle.

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Pythagorean Theorem - Find Missing Side - Whole Numbers

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Unknown Side Lengths in Right Triangles Worksheets

  • Geometry >
  • Pythagorean Theorem >
  • Finding Unknown Side Lengths

Upskill with our printable finding the unknown side lengths in right triangles worksheets! What is the length of the longest side of a right triangle whose legs are 8 units and 15 units in length? It's 17 units! We simply plugged the values in the Pythagorean equation and solved it. Now, what if the length of a leg goes missing? You're right! We rearrange the equation with the unknown length as the subject and press on as usual. To boil that down to formulas, we say c = √ (a 2 + b 2 ), a = √ (c 2 - b 2 ), b = √ (c 2 - a 2 ), where a and b are the leg lengths and c is the length of the hypotenuse. Everything will now go swimmingly when you obtain the missing side length of a right triangle!

Use our finding the missing side lengths in right triangles pdf worksheets for grade 7, grade 8, and high school.

Unknown Side Lengths in Right Triangles

Related Printable Worksheets

▶ Pythagorean Theorem Charts

▶ Finding the Hypotenuse

▶ Converse of the Pythagorean Theorem

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Right Triangle Side and Angle Calculator

How to find the sides of a right triangle, how to find the angle of a right triangle, how do you solve a right angle triangle with only one side, how to find the missing side of a right triangle how to find the angle example.

Finding the missing side or angle couldn't be easier than with our great tool – right triangle side and angle calculator. Choose two given values, type them into the calculator, and the calculator will determine the remaining unknowns in a blink of an eye! If you are wondering how to find the missing side of a right triangle, keep scrolling, and you'll find the formulas behind our calculator.

There are a few methods of obtaining right triangle side lengths. Depending on what is given, you can use different relationships or laws to find the missing side:

1. Given two sides

If you know two other sides of the right triangle, it's the easiest option; all you need to do is apply the Pythagorean theorem:

a² + b² = c²

If leg a is the missing side, then transform the equation to the form where a is on one side and take a square root:

a = √(c² - b²)

If leg b is unknown, then:

b = √(c² - a²)

For hypotenuse c missing, the formula is:

c = √(a² + b²)

🙋 Our Pythagorean theorem calculator will help you if you have any doubts at this point.

2. Given an angle and the hypotenuse

Right triangle with law of sines formulas. a over sin(α) equals b over sin(β) equals c, because sin(90°) = 1

Apply the law of sines or trigonometry to find the right triangle side lengths:

a = c × sin(α) or a = c × cos(β)

b = c × sin(β) or b = c × cos(α)

🙋 Refresh your knowledge with Omni's law of sines calculator !

3. Given an angle and one leg

Find the missing leg using trigonometric functions:

a = b × tan(α)

b = a × tan(β)

4. Given the area and one leg

As we remember from basic triangle area formula , we can calculate the area by multiplying the triangle height and base and dividing the result by two. A right triangle is a special case of a scalene triangle, in which one leg is the height when the second leg is the base, so the equation gets simplified to:

area = a × b / 2

For example, if we know only the right triangle area and the length of the leg a , we can derive the equation for the other sides:

  • b = 2 × area / a
  • c = √(a² + (2 × area / a)²)

🙋 For this type of problem, see also our area of a right triangle calculator .

If you know one angle apart from the right angle, the calculation of the third one is a piece of cake:

Given β : α = 90 - β

Given α : β = 90 - α

However, if only two sides of a triangle are given, finding the angles of a right triangle requires applying some basic trigonometric functions:

  • sin(α) = a / c so α = arcsin(a / c) (inverse sine);
  • cos(α) = b / c so α = arccos(b / c) (inverse cosine);
  • tan(α) = a / b so α = arctan(a / b) (inverse tangent);
  • cot(α) = b / a so α = arccot(b / a) (inverse cotangent);

and for β :

  • sin(β) = b / c so β = arcsin(b / c) (inverse sine);
  • cos(β) = a / c so β = arccos(a / c) (inverse cosine);
  • tan(β) = b / a so β = arctan(b / a) (inverse tangent);
  • cot(β) = a / b so β = arccot(a / b) (inverse cotangent).

To solve a triangle with one side, you also need one of the non-right angled angles . If not, it is impossible:

If you have the hypotenuse , multiply it by sin(θ) to get the length of the side opposite to the angle.

Alternatively, multiply the hypotenuse by cos(θ) to get the side adjacent to the angle.

If you have the non-hypotenuse side adjacent to the angle, divide it by cos(θ) to get the length of the hypotenuse .

Alternatively, multiply this length by tan(θ) to get the length of the side opposite to the angle.

If you have an angle and the side opposite to it, you can divide the side length by sin(θ) to get the hypotenuse .

Alternatively, divide the length by tan(θ) to get the length of the side adjacent to the angle.

Let's show how to find the sides of a right triangle with this tool:

Assume we want to find the missing side given area and one side. Select the proper option from a drop-down list . It's the third one.

Type in the given values . For example, the area of a right triangle is equal to 28 in² and b = 9 in.

Our right triangle side and angle calculator displays missing sides and angles! Now we know that:

  • a = 6.222 in
  • c = 10.941 in

Now, let's check how finding the angles of a right triangle works:

Refresh the calculator. Pick the option you need . Assume that we have two sides, and we want to find all angles. The default option is the right one.

Enter the side lengths . Our right triangle has a hypotenuse equal to 13 in and a leg a = 5 in.

Missing side and angles appear . In our example, b = 12 in, α = 22.62° and β = 67.38°.

How many lines of symmetry does a right triangle have?

If a right triangle is isosceles (i.e., its two non-hypotenuse sides are the same length), it has one line of symmetry . Otherwise, the triangle will have no lines of symmetry .

Can a right angled triangle have equal sides?

No, a right triangle cannot have all 3 sides equal , as all three angles cannot also be equal. One has to be 90° by definition. A right triangle can, however, have its two non-hypotenuse sides equal in length. This would also mean the two other angles are equal to 45°.

Are all right triangles similar?

Not all right-angled triangles are similar , although some can be. They are similar if all their angles are the same length, or if the ratio of two of their sides is the same.

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IMAGES

  1. 48 Pythagorean Theorem Worksheet with Answers [Word + PDF]

    pythagorean theorem worksheet find the missing side of each triangle

  2. Solved Pythagorean Theorem Sheet2 Determine the missing

    pythagorean theorem worksheet find the missing side of each triangle

  3. 48 Pythagorean Theorem Worksheet with Answers [Word + PDF]

    pythagorean theorem worksheet find the missing side of each triangle

  4. Answered: PYTHAGOREAN THEOREM

    pythagorean theorem worksheet find the missing side of each triangle

  5. Find the Missing Side of the Triangle Using the Pythagorean Theorem

    pythagorean theorem worksheet find the missing side of each triangle

  6. how to find a missing side in a right-angled triangle(Pythagoras' theorem)

    pythagorean theorem worksheet find the missing side of each triangle

VIDEO

  1. Pythagorean Theorem Worksheet #1

  2. PYTHAGOREAN THEOREM

  3. I solved (One solution of) the pythagorean theorem!

  4. Pythagorean Theorem Notes

  5. Geometry Ch7 Using Pythagorean Theorem, Find Missing Length and Area

  6. Algebra 1 The Pythagorean Theorem

COMMENTS

  1. PDF 8-The Pythagorean Theorem and Its Converse

    State if each triangle is acute, obtuse, or right. 15) Obtuse Acute 18) 4.8 km, 28.6 km, 29 km Obtuse Right Create your own worksheets like this one with Infinite Geometry. Free trial available at KutaSoftware.com

  2. Pythagorean Theorem Worksheets

    Apply Pythagorean theorem to find the unknown side of the right triangle. Round the answer to the nearest tenth. Download the set Unknown Length: Shapes Apply the Pythagorean theorem to find the unknown length of each shape in these printable worksheets. Round the answer to the nearest tenth. Download the set Word Problems | Level 1

  3. PDF Pythagorean Theorem Sheet 1

    Sheet 1 Determine the missing length in each right triangle using the Pythagorean theorem. Round the answer to the nearest tenth. 1) 2) A P ? B ? C Q 2 in R BC = PQ = 4) 5) E ? F X ? Z 14 yd Y G XZ = EF = 7) 8) S L 16 in N ? ? M T 1 cm U MN = ST = Printable Math Worksheets @ www.mathworksheets4kids.com 3) 6) 9) U 6 mm V ? W

  4. Pythagorean Theorem Worksheet: 4 Levels · Matter of Math

    Level 1: Identify the Right-Angle Triangles Level 2: Identify the Pythagorean Triples Level 3: Find the Missing Side Length of Right-Angle Triangles Level 4: Real-World Pythagorean Theorem Word Problems (2 Pages) The Man, the Legend: Pythagoras of Samos

  5. 48 Pythagorean Theorem Worksheet with Answers [Word + PDF]

    Following is an example that uses the Pythagorean Theorem to solve a triangle. a²+b²=c². 6²+8²=c². 36+64=c². 100=c². c²=100. c=√100. c=10. In this equation, the longest side of the triangle 'c' is missing. By finding out the sum of the squares of the two other sides, we were able to find the missing value.

  6. PDF The Pythagorean Theorem Date Period

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  7. Pythagorean Theorem Worksheets

    A worksheet where students use the Pythagorean Theorem to find the length of the missing side of right triangles. All triangle measurements are whole numbers and use Pythagorean triples. 6th through 8th Grades View PDF Pythagorean Theorem - No Illustrations

  8. PDF Right Triangle Trig Missing Sides and Angles

    Right Triangle Trig. - Finding Missing Sides and AnglesDate_____ Period____ Find the measure of each angle indicated. Round to the nearest tenth. 1) 13 12 B A C θ 2) 4 13 A B C θ 3) 9 6 A B C θ 4) 11.9 10 B A C θ 5) 7.7 14 A B C θ 6) 5 B 4 A C θ 7) 11 4.4 A B C θ 8) 3 3 B C A θ Find the measure of each side indicated. Round to the ...

  9. PDF Pythagorean Theorem Date Block

    Pythagorean Theorem Find the missing side of each triangle. Leave your answers in simplest radical form. 1) x 4 ft 15 ft 2) 7 mi 5 2 mi x 3) 11 cm 6 cm x 4) x 8 cm 16 cm 5) 10 in 4 in x 6) 2 in 7 in x State if each triangle is a right triangle. 7) 9 km 89 km 170 km 8) 2 26 yd 14 yd 15 yd 9) 4 mi 8 2 mi 12 mi 10) 5 km 11 km 6 km-1-

  10. PDF NAME: PYTHAGOREAN THEOREM

    NAME: PYTHAGOREAN THEOREM - WORKSHEET For each triangle find the missing length. Round your answer to the nearest tenth. For 5-8, side c is the hypotenuse of the right triangle ABC with sides a, b, c

  11. PDF PYTHAGOREAN THEOREM

    PYTHAGOREAN THEOREM - WORKSHEET. For each triangle find the missing length. Round your answer to the nearest tenth. Then find the area and the perimeter. 10. Find a third number so that the three numbers form a right triangle: 11. Ms. Green tells you that a right triangle has a hypotenuse of 13 and a leg of 5.

  12. PDF 7.3-7.4 Pythagorean Theorem

    Find the missing side of each triangle. Round your answers to the nearest tenth if necessary. Show your work throught the entire worksheet. 1) 12 m m 9 x 3) 12 km x 5 km 5) 7.9 ft 8.5 ft x 7) 10.4 in x 12 in 2) 5 mi 4 mi x 4) 10 mi x 6 mi 6) 11.8 cm x 8.9 cm m 8 8) x 8.2 m Worksheet by Kuta Software LLC

  13. PDF Missing Sides

    Title: missing-sides-1.pdf Author: USER 1 Created Date: 12/16/2023 2:47:57 PM

  14. Use Pythagorean theorem to find right triangle side lengths

    Use Pythagorean theorem to find right triangle side lengths. Google Classroom. Find the value of x in the triangle shown below. 6 8 x.

  15. Pythagorean Theorem

    30/09/2021 Country code: US Country: United States School subject: Math (1061955) Main content: Triangles (2012002) Find the missing side of a right triangle. Other contents: Pythagorean Theorem Share / Print Worksheet Finish! Find the missing side of a right triangle.

  16. PDF 7 Math Pythagorean Theorem Find the length of the missing side. 1. 2. 3

    Find the length of the missing side using the Pythagorean Theorem. 7. a = 7 and b = 24 8. b = 9 and c = 15 5 Pythagorean Theorem 5 40 9. a = 8 and c = 17 10. b = 3 and c = 5 11. a = 20and c = 21 12. a = 20 and b = 21 Each group of numbers represents the lengths of three sides of a triangle. Determine if the sides make a right triangle.

  17. Unknown Side Lengths in Right Triangles Worksheets

    You're right! We rearrange the equation with the unknown length as the subject and press on as usual. To boil that down to formulas, we say c = √ (a 2 + b 2 ), a = √ (c 2 - b 2 ), b = √ (c 2 - a 2 ), where a and b are the leg lengths and c is the length of the hypotenuse. Everything will now go swimmingly when you obtain the missing side ...

  18. How to Use Pythagoras' Theorem to Find Missing Sides on ...

    Example 2: Finding a Shorter Side. Now, take a look at the image below. This time, we have a right-angled triangle with one of the shorter sides missing. As it is a right-angled triangle, we still use Pythagoras' theorem, but we need to be careful where we substitute our numbers.

  19. Pythagorean Theorem Calculator

    Answer: hypotenuse c = area A = Get a Widget for this Calculator © Calculator Soup a = side leg a b = side leg b c = hypotenuse A = area What is the Pythagorean Theorem? The Pythagorean Theorem states that the sum of the squared sides of a right triangle equals the length of the hypotenuse squared.

  20. Pythagorean Theorem: Find the Missing Leg

    According to the Pythagorean theorem, the square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs, or a2 + b2 = c2. In this two-page geometry worksheet, students will practice using the Pythagorean theorem to find missing leg lengths on right triangles.

  21. Pythagorean Theorem Calculator

    This Pythagorean theorem calculator will calculate the length of any of the missing sides of a right triangle, provided you know the lengths of its other two sides. This includes calculating the hypotenuse. The hypotenuse of the right triangle is the side opposite the right angle, and is the longest side.

  22. Pythagoras' theorem

    A worksheet where you need to use Pythagoras' theorem to find the hypotenuse of a right-angled triangle. Choose if you want the problems to be in metric units or imperial units. Number of problems 4 problems 8 problems

  23. Right Triangle Calculator

    b = √ (c² - a²) For hypotenuse c missing, the formula is: c = √ (a² + b²) 🙋 Our Pythagorean theorem calculator will help you if you have any doubts at this point. 2. Given an angle and the hypotenuse. Apply the law of sines or trigonometry to find the right triangle side lengths: a = c × sin (α) or a = c × cos (β)