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.

Pythagoras Theorem in 3D Shapes

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.

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.

Pythagoras Theorem in 3D Shapes

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.

Pythagoras Theorem in 3D Shapes

Leave a Reply

Your email address will not be published. Required fields are marked *

You may use these HTML tags and attributes:

<a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <s> <strike> <strong>

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Mr Mathematics Blog

Sharing an Amount to a Given Ratio

There are three common methods for sharing an amount to a given ratio.  Depending on the age group and ability range I am teaching I would choose one approach over the other two. The three methods are: Using fractions Unitary method Using a table In this blog I will demonstrate each of the three methods […]

Plotting Scatter Graphs and Understanding Correlation

To introduce plotting scatter graphs and understanding correlation I ask students to think about the relationships between different variables and to describe how they might be related. Here’s my starter activity which students discuss in pairs then present to me on mini-whiteboards. When the students have had time to discuss the matching pairs we talk […]

Plotting Quadratic Graphs on Cartesian Axes

In recent examiner reports it is noted how important it is for students to understand the properties of a parabola  when plotting quadratic graphs on Cartesian axes.  Students who have a secure understanding of parabolas can use them to correct miscalculated values in their table of results and are more likely to attain full marks […]