What is the theorem of 30 60 90
What is a 30-60-90 triangle?
|......||If you halve an equilateral triangle by a height, you get a triangle with interior angles of 30 °, 60 ° and 90 °.|
That is why it is also called the 30-60-90 triangle.
When a triangle is mentioned on this page, it is usually the 30-60-90 triangle.
Sizes of the triangle Top
|......||If the hypotenuse of the exit triangle is the same cso are the cathets b= (1/2) c and a= (1/2) sqr (3) c. The following applies to the aspect ratio:b:c = sqrt (3):1:2.|
The area is A.= (1/2) ab = (1/8) sqr (3) c².
|......||Since the 30-60-90 triangle is right-angled, there is only one height h, the other heights coincide with the cathetus. The height divides the hypotenuse c into the hypotenuse sections p and q.|
and p= (3/4) c, q= (1/4) c [follows from the cathetus set pc = b² or qc = a²].
|Like every triangle, the 30-60-90 triangle has a perimeter and an inscribed circle.|
Half of the circumference is the circle of the valley with the radius R.= c / 2.
The radius of the inscribed circle is r= (1/4) [3 - sqr (3)] c.
|......||Look at the blue triangle. The following applies: tan (30 °) = r / (a-r). If one sets tan (30 °) = 1 / sqr (3) and a = c / 2, after some transformations r = (1/4) [3 - sqr (3)] c results.|
Squares in the triangle Top
|There are two ways to put a square inside a 30-60-90 triangle.|
Are the squares the same size?
|......||To solve this, the triangle is placed in a coordinate system and the straight line g is considered1 and G2. |
The straight line g1 : f (x) = - (b / a) * x + b contains the hypotenuse,
the straight line g2 : g (x) = x is the 1st bisector and contains two corners of the square.
For the intersection of the two straight lines, f (x) = g (x) or x = -b / a * x + b or applies x = ab / (a + b).
ax = -bx + ab | + bx
ax + bx = ab | Exclude x
(a + b) x = ab |: (a + b)
x = ab / (a + b)
|According to the 2nd theorem of rays (slightly modified), h: c = (h-x): x applies. It follows x = hc / (h + c).|
Incidentally, the statements apply to any right-angled triangles.
If you insert the sizes of the 30-60-90 triangle, the result is
in the first case x = (1/4) [3-sqr (3)] c (about 0.32c),
in the second case x = (1/13) [4sqr (3) -3] c (about 0.30c).
Figures made from 30-60-90 triangles Top
Squares, equilateral triangles or isosceles right-angled triangles can be put together in such a way that new figures are created. They are then called polyominos (pentominos or hexominos), Polyiamonds or polybolos.
Of course, you can also create new figures from 30-60-90 triangles.
|You can put two triangles together in six different ways. They are not well suited as puzzle pieces because all routes are different from each other.|
However: These figures appear in the Eternity game and are called Polydrafter (see link list).
Figures made up of six triangles
|If you draw all heights in an equilateral triangle, you get six 30-60-90 triangles. You can use them as tangram stones. (1)|
A figure made up of eight triangles
|......||If you draw the diagonals and the center lines in a rectangle, you get eight right-angled triangles.|
The figure corresponds to the English national flag. In general, the triangles are not 30-60-90 triangles. This only applies if the flag has the format sqr (3): 1.
|......||Six straight lines meet at one point and form 12 angles of size 30 °.|
If one specifies the vertical line of length a and continues it in the angular spaces by drawing a perpendicular to the next half-line, a sequence of ever smaller 30-60-90 triangles is created. The shorter cathets form a spiral (red).
What limit does the length of the spiral approach if a is given?
|Given is a.|
The sequence of hypotenuses b1, b2, b3, b4, ... form a geometric sequence.
b2 = [(1/2) sqr (3)] (2a)
b3 = [(1/2) sqr (3)] b2 = [(1/2) sqr (3)] ² (2a)
b4 = [(1/2) sqr (3)] b3 = [(1/2) sqr (3)] ³ (2a)
The following applies to the shorter catheters that form the spiral:
a1 = a
a2 = (1/2) b2 = [(1/2) sqr (3)] a
a3 = (1/2) b3 = [(1/2) sqr (3)] ²a
a4 = (1/2) b4 = [(1/2) sqr (3)] 3 a
That is a geometric sequence. The associated series has the limit value 1 / (1-q) = [4 + 2sqr (3)] a
Two more figures
Triangles in a 30-60-90 triangle
A star made up of nine triangles
Two cones Top
|Two cones are created when the triangle rotates around one of the two cathets.|
Let V (a) be the volume when rotating around a, V (b) around b.
The volumes of the two cones are in the ratio V (b): V (a) = sqr (3): 1.
The triangle can also rotate around the hypotenuse. Then a double cone arises with V (q): V (p) = 3: 1.
Eternity puzzle Top
|......||......||Puzzles are known in which one should put together a rectangular picture from individual pieces.|
In this case there are 24 pieces.
In June 1999 the British company Racing Champions Ltd brought the so-called Eternity Puzzle onto the market. The inventor was Christopher Monckton. According to (2) the puzzle has been sold more than 250,000 times.
|The puzzle consists of 209 pieces. Each part is made up of twelve 30-60-90 triangles.|
With these parts an almost regular dodecagon should be laid out.
If a is the length of the side and h is the height of the equilateral triangle, the dodecagon has alternating sides 7a and 8h.
You can find more via my link list.
30-60-90 triangle on the internet Top
Eric W. Weisstein (MathWorld)
30-60-90 Triangle, Eternity, Polydrafter
Ed Pegg Jr. (Math Puzzles)
THE ETERNITY PUZZLE
30 ° - 60 ° - 90 ° triangle
Lawrence Spector (TheMathPage)
THE 30 ° -60 ° -90 ° TRIANGLE
30-60-90 triangle, Eternity puzzle
(1) Karl-Heinz Koch: ... lege Spiele, Cologne 1987 (ISBN 3-7701-2097-3)
(2) Ingo Althöfer: One million British pounds for two mathematicians, Omega magazine, Spektrum Spezial 4/2003
URL of my homepage:
© 2003 Jürgen Köller
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