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Stage 5.1-3

10.10 Pythagoras in 3D

Lesson

Introduction

Pythagoras' theorem gives us a relationship between the three sides of a right-angled triangle, allowing us to use any two sides to find the third. This theorem can be used for any right-angled triangle, even those in a three dimensional context.

Lengths in 3D space

We can identify right-angled triangles in 3D space the same way that we do in 2D space, by looking for a right angle.

In 3D space, right angles occur between lines that are perpendicular in the same 2D plane. In other words, if two lines meet at a right-angle in some 2D slice of 3D space, the triangle formed with these two lines is a right-angled triangle.

A cube with square base A B C D and square top G H E F.

Consider this cube with its vertices labelled.

Square A B C D split along diagonal D B into 2 right angled triangles.

If we take the 2D slice through the vertices A, B, C and D we can see that any triangle determined by three of these vertices will be right-angled, since this slice gives us a square.

In fact, any of the 2D slices that corresponds to a face of the cube will give us multiple possible right-angled triangles.

Rectangle D B H F split along diagonal F B into 2 right angled triangles.

Taking a 2D slice through the vertices F, \,H,\,B and D gives us a rectangle, which means that we still form a right-angled triangle from any three of these vertices.

A cube with square base A B C D and square top G H E F. Right angled triangles D B F and A B D are inside the cube.

If we return to 3D space, we can see how these right-angled triangles look in 3D.

We can perform a similar exercise with other 3D solids like cones and prisms to find other right-angled triangles in 3D space:

A cone and a triangular pyramid with right-angled triangles inside them. Ask your teacher for more information.

In most straight-edged solids, the diagonals formed by joining two non-adjacent vertices will often be the hypotenuse of some right-angled triangle in that solid.

In order to find the length of such a diagonal, we can simply use Pythagoras' theorem to calculate the hypotenuse, provided we know the lengths of the other two sides.

In the cases where we don't know both the other side lengths, we may need to use Pythagoras' theorem on a different right-angled triangle in the solid, as seen in the example below.

Examples

Example 1

A square prism has sides of length 9\,cm, 9\,cm and 16\,cm as shown.

A rectangular prism with dimensions 9, 9, and 16 centimetres. Ask your teacher for more information.
a

If the diagonal HF has a length of z\,cm, calculate the exact value of z, leaving your answer in surd form.

Worked Solution
Create a strategy

We can use Pythagoras' theorem to find the length of HF in the triangle \triangle FGH.

Apply the idea
\displaystyle HF^2\displaystyle =\displaystyle HG^2+GF^2Use the Pythagoras' theorem
\displaystyle z^2\displaystyle =\displaystyle 9^2+16^2Substitute the lengths
\displaystyle z^2\displaystyle =\displaystyle 337Evaluate the right side
\displaystyle z\displaystyle =\displaystyle \sqrt{337}Take the square root of both sides
b

Find y, the length of the diagonal DF to two decimal places.

Worked Solution
Create a strategy

We can use Pythagoras' theorem to find the length of DF in the triangle \triangle DHF.

Apply the idea

Consider the right-angled \triangle FDH. The hypotenuse can be found using Pythagoras' Theorem, c^2=a^2+b^2.

\displaystyle DF^2\displaystyle =\displaystyle DH^2+HF^2Use the Pythagoras' theorem
\displaystyle y^2\displaystyle =\displaystyle 9^2+(\sqrt{337})^2Substitute the lengths
\displaystyle y^2\displaystyle =\displaystyle 81+337Evaluate the powers
\displaystyle y^2\displaystyle =\displaystyle 418Evaluate the addition
\displaystyle y\displaystyle =\displaystyle \sqrt{418}Take the square root of both sides
\displaystyle =\displaystyle 20.45Evaluate

Example 2

A soft drink can has a height of 13\,cm and a radius of 5\,cm. Find L, the length of the longest straw that can fit into the can (so that the straw is not bent and fits entirely inside the can).

A cylinder with radius of 5 centimetres, height of 13 centimetres, and diagonal length of L.

Round your answer down to the nearest centimetre, to ensure it fits inside the can.

Worked Solution
Create a strategy

Use the Pythagoras' theorem to find the length L using the height and radius of the can.

Apply the idea

L is the hypotenuse of the right-angled triangle. The other sides are the height which is 13 and the diameter of the can which is 2 \times 5=10.

\displaystyle c^2\displaystyle =\displaystyle a^2+b^2Use the Pythagoras' theorem
\displaystyle L^2\displaystyle =\displaystyle 13^2+(10)^2Substitutes L, the height and diameter
\displaystyle L^2\displaystyle =\displaystyle 269Evaluate the right side
\displaystyle L\displaystyle =\displaystyle \sqrt{269}Take the square root of both sides
\displaystyle \approx\displaystyle 16.4Evaluate

The length of the longest straw to the nearest centimetre is 16 cm.

Idea summary

We can find right-angled triangles in 3D solid by taking 2D slices. Once we identify right-angled triangles we can use Pythagoras' theorem to find unknown lengths in the solids.

Outcomes

MA5.3-15MG

applies Pythagoras' theorem, trigonometric relationships, the sine rule, the cosine rule and the area rule to solve problems, including problems involving three dimensions

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