The document discusses applying the Pythagorean theorem in three dimensions to determine the longest stick that can fit in a box. It explains that the length of the stick (c) is equal to the sum of the squares of the box's height (a), width (d), and depth (e), or c = a^2 + d^2 + e^2, providing a shortcut to calculate the longest stick without finding intermediate dimensions first.

1. Pythagoras in
3 Dimensions

2. What’s the longest stick you
can put in the box?

3. What’s the longest stick you
can put in the box?

4. What’s the longest stick you
can put in the box?

5. What’s the longest stick you
can put in the box?

6. What’s the longest stick you
can put in the box?

7. What’s the longest stick you
can put in the box?
𝑎
𝑏
𝑐
𝑐 = 𝑎2 + 𝑏2
…but we don’t
know 𝑏

8. What’s the longest stick you
can put in the box?
𝑏 = 𝑑2 + 𝑒2
𝑏
𝑑
𝑒

9. What’s the longest stick you
can put in the box?
𝑏 = 𝑑2 + 𝑒2
so find 𝑏 first,
then find 𝑐
𝑑
𝑒
𝑏

10. What’s the longest stick you
can put in the box?
𝑏 = 𝑑2 + 𝑒2
so find 𝑏 first,
then find 𝑐
OR
the shortcut:
𝑏
𝑑
𝑒

11. What’s the longest stick you
can put in the box?
𝑏 = 𝑑2 + 𝑒2
or
𝒃 𝟐 = 𝒅 𝟐 + 𝒆 𝟐
𝑏
𝑑
𝑒

12. What’s the longest stick you
can put in the box?
𝑏
𝑑
𝑒
𝑎
𝑐 = 𝑎2 + 𝑏2
…and
𝒃 𝟐
= 𝒅 𝟐
+ 𝒆 𝟐
so
𝑐 = 𝑎2 + 𝒅 𝟐 + 𝒆 𝟐
𝑐

13. What’s the longest stick you
can put in the box?
𝒅
𝒆
𝒂
Shortcut:
𝑐 = 𝒂2 + 𝒅 𝟐 + 𝒆 𝟐
𝑐

14. What’s the longest stick you
can put in the box?
𝒅
𝒆
𝒂
Shortcut:
𝑐 = 𝒂2 + 𝒅 𝟐 + 𝒆 𝟐
𝑐 = 𝒉𝒆𝒊𝒈𝒉𝒕2 + 𝒘𝒊𝒅𝒕𝒉 𝟐 + 𝒅𝒆𝒑𝒕𝒉 𝟐
𝑐

15. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0
International License.