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.
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.
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.
The interactive Excel file can be used for additional practise before students work independently through the questions on the third slide and worksheet.
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.
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.
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 […]
Drawing frequency trees for GCSE maths is a new topic and appears on both the higher and foundation curriculum. I’ve taught this lesson a couple of times, once to Year 10 and once to Year 11 and I have to say the kids really enjoy it. What is a frequency tree? Frequency trees can be […]
Whenever I teach how to calculate speed as a measure of distance and time I either use the formula or the triangle method. In my experience most students are know about the triangle method from their science lessons. For this reason I would have expected speed to appear either within the algebra or shape and measures strands of […]