This document discusses 30-60-90 triangles and provides shortcuts for determining side lengths. It states that in a 30-60-90 triangle: (1) the side opposite the 30 degree angle is the short side s, the side opposite the 60 degree angle is the long side l, and the hypotenuse is h; (2) the relationship between s and h is h = 2s; and (3) the relationship between s and l is l = s√3. It then demonstrates using these shortcuts to find side lengths when given s, l, or h through worked examples.

1. Special Right
Triangles
30 – 60 – 90 Triangles

2. Special Right Triangles
Directions
As you view this presentation, take
notes and work out the practice
problems.
When you get to the practice problem
screens, complete the step in your
notebook before continuing to the
next slide.

3. 30- 60- 90 Triangles
l
s
30o
60o
h
• In a 30 – 60 – 90 triangle, the
side across from the 30o angle is
the short side and often labeled
s.
• In a 30 – 60 – 90 triangle, the
side across from the 60o angle is
the long side and often labeled
l.
• The hypotenuse is often labeled
h.

4. 30- 60- 90 Triangles
Understanding the Shortcuts
s
30o
60o
h
l
To understand the
relationship between the
short side and the
hypotenuse, draw a second
30 - 60 – 90 triangle with
the same dimensions as the
original triangle. Arrange
the triangles to form an
equilateral triangle with
side l as the common side.

5. 30- 60- 90 Triangles
UnderstandingtheShortcut forFindingtheLength
ofthe Hypotenuse
s
30o
60o
h
l
h
s
Because the triangle
is an equilateral
triangle, s + s = h or
2s = h

6. 30- 60- 90 Triangles
UnderstandingtheShortcutforFindingtheLengthof
theLongLeg
s
30o
60o
h = 2s
l
h
s
The Pythagorean
Theorem is used to
show the relationship
between the long
side, l, the short side,
s, and the
hypotenuse, h.
s2 + l2 = h2
s2 + l2 = (2s)2
l2 = 4s2 – s2
l2 = 3s2
l2 = 3s2
l = s 3

7. 30- 60- 90 Triangles
UsingtheShortcutswhens isKnown
s
30o
60o
h = 2sl = s 3
When the Short Side
is known:
Short side = s
Long side = s 3
Hypotenuse = 2s

8. 30- 60- 90 Triangles
Practice Problem 1
s = 5
30o
60o
Finding the lengths
of the hypotenuse
and long side when
s = 5 l = ? h= ?

9. 30- 60- 90 Triangles
Practice Problem 1
s = 5
30o
60o
h = 2s
l = s 3
Finding the lengths of the
hypotenuse and long side
when
s = 5
Remember the shortcuts

10. 30- 60- 90 Triangles
Practice Problem 1
30o
60o
h = 10l = 5 3
l = s 3 = 5 3
h = 2s = 2* 5 = 10
Finding the lengths of the
hypotenuse and long side
s = 5
Remember the shortcuts
s = 5

11. 30- 60- 90 Triangles
UsingtheShortcutswhenlisKnown
s =
𝑙 3
3
30o
60o
h =
2𝑙 3
3
l
Long Side = l
Short Side
l = s 3
l/ 3= s 3 / 3
𝑙 3
3
= s
Hypotenuse
h = 2s
OR
h = 2(
𝑙 3
3
)
h =
2𝑙 3
3

12. 30- 60- 90 Triangles
Practice Problem 2
30o
60o
Finding the lengths
of the hypotenuse
and short side when
l = 7 l = 7 h = ?
s = ?

13. 30- 60- 90 Triangles
Practice Problem 2
30o
60o
Finding the lengths
of the hypotenuse
and short side when
l = 7
Remember the
shortcuts
l = 7
s =
𝑙 3
3
h = 2s =
=
2𝑙 3
3

14. 30- 60- 90 Triangles
Practice Problem 2
30o
60o
Finding the lengths
of the hypotenuse
and short side when
l = 7
Remember the
shortcuts
s =
𝑙 3
3
=
7 3
3
h = 2s = 2(
7 3
3
)=
14 3
3
l = 7
s =
7 3
3
h =
14 3
3

15. 30- 60- 90 Triangles
UsingtheShortcutswhenh isKnown
s = h/2
30o
60o
h
l =
h 3
2
Hypotenuse = h
Short Side
h = 2s
h/2 = 2s/2
h/2 = s
Long Side
l = s 3
OR
l = (h/2) 3 =
h 3
2

16. 30- 60- 90 Triangles
Practice Problem 3
30o
60o
Finding the lengths
of the short side and
the long side when
h = 1
h = 1
s = ?
l = ?

17. s = h/2
30o
60o
h = 1
l =
h 3
2
Finding the lengths of
the short side and the
long side when
h = 1
Remember the
shortcuts
30- 60- 90 Triangles
Practice Problem 3

18. s =
1
2
30o
60o
h = 1
l =
3
2
Finding the lengths of
the short side and the
long side when
h = 1
Remember the
shortcuts
s =
ℎ
2
=
1
2
l = s 3 = (
1
2
) 3 =
3
2
30- 60- 90 Triangles
Practice Problem 3

19. s =
1
2
30o
60o
h = 1l =
3
2
In the Unit Circle:
h = 1
So remembering these
shortcuts for the 30 – 60 – 90
triangle will save you time and
work.
s =
1
2
l =
3
2
30- 60- 90 Triangles
in the Unit Circle