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      Lesson details

      Learning outcome

      I can use Pythagoras' theorem to find the length of the hypotenuse.

      Key learning points

      1. The sum of the squares of the two shorter sides equals the square of the longest side.
      2. The longest side is always opposite the right angle.
      3. A calculator can perform these calculations efficiently.
      4. Priority of operations makes the order clear.

      Keywords

      • Pythagoras’ theorem - Pythagoras’ theorem shows that the sum of the squares of the two shorter sides of a right-angled triangle is equal to the square of its longest side (the hypotenuse).

      • Hypotenuse - A hypotenuse is the side of the right-angled triangle which is opposite the right-angle.

      • Right-angled triangle - A right-angled triangle has exactly one 90° interior angle.

      Common misconception

      Pythagorean triples can be a trio of any rational numbers that, when constructed into a triangle, always produces a right-angled triangle.

      Pythagorean triples are conventionally a trio of integer side lengths of a right-angled triangle, such as the 3, 4, 5 triangle. Other, similar triangles can be generated from Pythagorean triples, whose side lengths are rational, such as 0.3, 0.4, 0.5

      Teacher tip

      Whilst it is encouraged for students to transition to more algebraic methods, it is still okay at this stage for students to rely on drawing on a square from each side of the triangle when using Pythagoras' theorem.

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
      except where otherwise stated. See Oak's terms & conditions
      (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

      6 Questions

      Q1.
      Which of these sides is a hypotenuse?

      An image in a quiz
      side A
      Correct answer: side B
      side C
      none of them are hypotenuses
      it is impossible to tell

      Q2.
      The triangle formed from these three squares is right-angled. What is the value of $$n$$, where $$n$$ units² is the area of a square.

      An image in a quiz
      2
      4.9
      12
      Correct answer: 24
      290

      Q3.
      If three squares with different areas are joined at their vertices, what type of triangle would be formed?

      acute triangle
      Correct answer: scalene triangle
      isosceles triangle
      equilateral triangle

      Q4.
      Angle $$a°$$ is an acute angle, but is the largest angle in this triangle. Which of these are possible values of $$t$$, where $$t$$ units² is the area of a square.

      An image in a quiz
      17
      Correct answer: 19
      Correct answer: 20
      32
      34

      Q5.
      Which of these are true for the Pythagoras' theorem?

      Describes a property of only equilateral triangles.
      Correct answer: Describes a property of only right-angled triangles.
      Correct answer: Describes a relationship between the squares of the side lengths of a triangle.
      Describes a relationship between the interior angles of a triangle.
      Correct answer: Describes a relationship between squares formed from the sides of a triangle.

      Q6.
      A right-angled triangle is formed from three squares. The area of two of the squares are 75 units² and 25 units². What are the possible areas of the third square?

      3 units²
      25 units²
      Correct answer: 50 units²
      Correct answer: 100 units²
      6250 units²

      6 Questions

      Q1.
      The length of the hypotenuse for this right-angled triangle is cm.

      An image in a quiz
      Correct Answer: 11 cm, 11cm

      Q2.
      $$k$$ cm² is the area of the square from the hypotenuse of the triangle. The value of $$k$$ is .

      An image in a quiz
      Correct Answer: 67.24, 67.24 cm², 67.24cm², 67.24 cm squared, 67.24cm squared

      Q3.
      Calculate the length of the hypotenuse of this triangle, in units.

      An image in a quiz
      10 units
      12.69 units
      14.14 units
      Correct answer: 19 units
      361 units

      Q4.
      The area, $$w$$, of the largest square in this diagram is cm².

      An image in a quiz
      Correct Answer: 2664, 2664 cm², 2664cm², 2 664, 2,664

      Q5.
      Calculate the length of the hypotenuse for this triangle. (Give your answer to 1 d.p.).

      An image in a quiz
      Correct answer: 24.1 units²
      506.8 units²
      583 units²
      256 889 units²

      Q6.
      The length of the hypotenuse of this triangle, rounded to 1 d.p. is cm.

      An image in a quiz
      Correct Answer: 23.4

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