NZ Level 6 (NZC) Level 1 (NCEA)
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PYTHAG - Applications using Pythagoras

Remember Pythagoras' Theorem?

Pythagoras' Theorem

$a^2+b^2=c^2$a2+b2=c2, where

  • $c$c is the length of the hypotenuse, and
  • $a$a and $b$b are the lengths of the other two sides

We can rearrange this equation to find formulas for each side length.

Rearranging Pythagoras' Theorem

To find the hypotenuse:        $c=\sqrt{a^2+b^2}$c=a2+b2

 To find a shorter side:     $a=\sqrt{c^2-b^2}$a=c2b2

To apply Pythagoras' Theorem to real life situations,

  1. Look for right-angled triangles
  2. Choose which side, hypotenuse or a shorter side, you are trying to find
  3. Find the lengths of the other two sides
  4. Apply the relevant formula and substitute the lengths of the other two sides

Let's look at some examples so we can see this in action.


Worked Examples

Question 1

Consider a cone with slant height $13$13m and perpendicular height $12$12m.

  1. Find the length of the radius, $r$r, of the base of this cone.

  2. Hence, find the length of the diameter of the cone's base.

Question 2


Find the length of the unknown side, x, in the given trapezium.

Give your answer correct to $2$2 decimal places.




Use trigonometric ratios and Pythagoras’ theorem in two and three dimensions


Apply geometric reasoning in solving problems


Apply right-angled triangles in solving measurement problems

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