The 22.5-67.5-90 Triangle

22p5-67p5-90-triangle corrrected

In the diagram above, a regular octagon is shown nested inside square LMNP. The central angles of this octagon, such as angle HAF, each measure 360/8 = 45 degrees. Segments HA and FA are radii, and G is the midpoint of HF, making GF a half-side and GA an apothem. Since this apothem bisects angle HAF, angle GAF is 22.5 degrees, making the yellow triangle a 22.5-67.5-90 triangle.

Let FH = 2, as well as FK (and the other six sides of the regular octagon, as well), and GF would then equal 1, since G is the midpoint of FH. Triangle KNF is a 45-45-90 triangle with hypotenuse length 2, giving it a leg length of 2/√2, or simply √2. This makes segment XN (with X the midpoint of EK) have a length of 1 + √2, and the light blue segment, AG, has this same length of 1 + √2, by horizontal translation to the left.

The hypotenuse of the yellow 22.5-67.5-90 triangle can then be found using the Pythagorean theorem, since it is is known that the short leg (GF) has a length of 1, while the long leg (AG) has a length of 1+√2. Let this hypotenuse (AF, shown in red) be x, and then x2 = 12 + (1 + √2)2 = 1 + 1 + 2√2 + 2 = 4 + 2√2, so x, and therefore the hypotenuse, has a length of √(4+2√2).

The 22.5-67.5-90 triangle, therefore, has a short leg:long leg:hypotenuse ratio of 1:1+√2:√(4+2√2).

About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things. The majority of these things are geometrical. Welcome to my little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with the views of my employer, nor any other organization, nor institution, of any kind.
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5 Responses to The 22.5-67.5-90 Triangle

  1. Xioneks says:

    In the second paragraph
    “Let FN =2 …”
    Did i mean FK instead?

    Like

  2. Xioneks says:

    *Did you meant FK instead?
    (Typo error)

    Like

  3. Yes, and thank you. Good eye! I will correct the error.

    Like

  4. 5:12:13 is an amazingly good approximation to this

    Liked by 1 person

  5. Anonymous says:

    Thanks for your help
    I need this for an stellar equation I’m making that may replace pothagorians therom.

    Like

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