Pythagoras’ Theorem

Introducing Pythagoras’ Theorem

Finding the Hypotenuse

Finding the Shorter Side

Pythagoras’ Theorem in 3D

Applying Pythagoras’ Theorem

Prerequisite Knowledge

• derive and apply the properties and definitions of: special types of quadrilaterals, including square, rectangle, parallelogram, trapezium, kite and rhombus; and triangles and other plane figures using appropriate language
• calculate the perimeters of 2D shapes, including composite shapes;
• use the standard conventions for labelling and referring to the sides and angles of triangles; draw diagrams from written description

Success Criteria

• know the formulae for: Pythagoras’ theorem, a² + b² = c²
• apply angle facts, triangle congruence, similarity and properties of quadrilaterals to conjecture and derive results about angles and sides, including Pythagoras’ Theorem and the fact that the
• base angles of an isosceles triangle are equal, and use known results to obtain simple proofs

Key Concepts

• Pythagoras’ Theorem identifies how the three sides of a right angled triangle are connected by the areas of shapes on each edge. To fully engage with this concept students could construct the theorem using a 3,4,5 triangle to measure the hypotenuse and calculate the area of each square. Their hypothesis can then by tested on a 5, 12, 13 triangle.
• Pythagoras’ Theorem can be applied to a wide variety of geometrical and real world problems. Students need to practise identifying when the theorem can be applied by recognising triangular components.

Common Misconceptions

• Students often believe that the areas of the shapes on the edges have to be squares in order for a² + b² = c² to apply. In fact, the formula applies for all shapes as long as the dimensions are in proportion to the edges of the triangle.
• Confusion often lies in identifying the Hypotenuse side of a right-angled triangle since it is not always apparent which side is longest. Encourage students to identify the hypotenuse as opposite the right angle.
• There is often difficulty when trying to calculate a shorter side of a triangle since students tend to memorise the formula with the hypotenuse as the subject.