# Pythagoras Theorem in 3D Shapes

Many problems involve three-dimensional objects or spaces.  Pythagoras Theorem in 3D Shapes can be used as much with these problems as those in plane shapes.

### Knowing when to use Pythagoras Theorem

The starter recaps applying Pythagoras Theorem as part of a larger problem involving the perimeter of a trapezium and square.  The aim of this starter is for students to experience the range of problems that can be solved using the theorem.  In this instance there are no obvious right-angled triangles yet the problem cannot be solved without Pythagoras.

### Pythagoras Theorem in 3D Shapes

When you work with 3D shapes it is important to look for and draw out right-angled triangles.  Labelling the vertices of the triangle with the same label as those used in the 3D shape helps students to visualise it two dimensionally.

The base ABCG is in a horizontal plane and triangle ACD is in a vertical plane.  First look at the right-angled triangle ABC and use Pythagoras’ Theorem to find length AC.

Next, look at triangle ACD.  Label the length of AC that was found in triangle ABC, leave the answer in its most exact form as the root will be cancelled when squared. Use Pythagoras’ Theorem to work out length AD. We work through the second problem as a class.  I encourage students to draw the cuboid on one half of their mini-whiteboard and in the second half draw the right-angled triangles.  Similar to the diagram above.  It’s important to leave the working of the first example on the board to help students see the various stages.  Click here to watch the video.

### Further Practice

The interactive Excel file can be used for additional practise before students work independently through the questions on the third slide and worksheet.

### Pythagoras Theorem in Composite Solids

In the plenary students are challenged to find the total perpendicular height of a composite solid.  In addition to this being a composite shape a new level of difficulty is added as students now have to use the hypotenuse to find a shorter side. ### Extended Learning

I like to set the problem shown below for homework after this lesson as it challenges students to link Pythagoras’ Theorem with volume of pyramids.

The diagram shows a pyramid.

ABCD is a square with lengths 8 cm.

The other faces of the pyramid are equilateral triangles with sides of length 8 cm.

Calculate the volume of the pyramid. ## Teach this lesson

### Solving 3D Problems using Trigonometry

When solving 3D problems involving missing lengths and angles it is important to draw 2D right-angled-triangles to apply the Sine, Cosine and Tangent trigonometric ratios.

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