Calculate Area of the Green shaded Triangle | Important Geometry skills explained

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Calculate Area of the Green shaded Triangle | Important Geometry skills explained

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Calculate Area of the Green shaded Triangle

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[יוסיבוקר-דו]

Very elegant solution. However, you need to prove that all small triangles are equal in size. It might seem obvious, but this is not an axiom or a well known formula, so it requires proofing. I solved it in a much longer way. From the exagon space formula you generate the exagon's side length. If you extend the exagon's adjacent latetals until they meet, you get an equilateral triangle. This allows you to calculate the inner triangle lateral length, based on similar triangles ratio. It is equal to 3/2 the length of the exagon lateral. Once you know it, you can easily calculate the inner triangle space based on the formula for equilateral triangles.

[jaycewheeled]

Very nice and elegant solution. It took me a little bit longer by demonstrating first that the side of the triangle is 3/2 the side of the hexagon. From there it became very easy

[bigm]

I was ready for a wild trigonometric ride and ended up with little triangles instead!👍😂😂

[davidfromstow]

Extremely interesting problem and solution - thank you

[harikatragadda]

Extending the sides (h) of the Hexagon forms an Equilateral Triangle of side length equal to the side of the Green Triangle(L).
But this Triangle also has a length = h+½h =3h/2
Therefore L=3h/2
Since the Hexagon is made of six Equilateral triangles of side h,
each with Area (A) =
(Area of Hexagon)/6 = 10
A=10
Area of Green Triangle=(3/2)²A =9A/4 = 22.5

[plamenpenchev]

Without pen and paper.
Six equilateral triangles in hexagon with area 10 and base a. The middle hecagon's horizontal line is 2a. (Very easy to prove) In lower trapezoid green side is in the middle and its length is (a + 2a)/2 = 3/2*a.
Then green area is (3/2)^2×10 = 45/2.

[spiderjump]

You can extend the lines at the vertices and get an equilateral triangle with side length of hexagon which has an area of 60/6 =10 cm^2. By similar triangles,
The green triangle has side length 1.5 times longer than the smaller equilateral triangle so the area will be 10•1.5•1.5= 22.5

[Waldlaeufer]

A regular hexagon consists of six regular triangles.
Let the side of a single triangle be 2. Then its height is √3 and its area is 1/2 * 2√3 = √3.
Six of these triangles cover an area of 6√3.
In this scale, the side length of the green regular triangle is 3 and its area therefore is 1/2 * 3 * 3/2 *√3 = 9/4 √3.
This is 9/4 √3 /(6√3) = 9/24 = 3/8 of the hexagon.
So the area of the green triangle is 3/8 of 60, which is 22.5.

P.S. In fact, I ended up counting the little triangles to make sure my calculations were correct. 🤪

[johnnyragadoo]

I stopped the video after the problem description and had a go at a solution. Here's what I came up with, rather lazily using the sin and cos buttons on my calculator.

If the area of a regular hexagon is 60, the sides are sqrt(60/3cos(30)), or 4.81. The sloped sides of the outer trapezoids, the three trapezoids outside the green triangle, are half of that, or about 2.4.

If we construct perpendiculars to chop those outer trapezoids into rectangles with "bookends" of two right triangles, we can subtract degrees, consider the angles around parallel lines, and determine the two outer triangles are right triangles with angles 30, 60, and 90 degrees.

The base of the rectangle is the length of a side of the hexagon, or 4.81. The height is side/2 * cos(30), or about 2.08. The area of the rectangle is 4.81 * 2.08. Using full precision, that comes to 10. There are three of those rectangles, so there are 30 square units in them.

The little triangles are right triangles with one leg 2.08 and the other leg (side/2)*sin(30), or about 1.2.

Quick cross check - from pythagoras, sqrt(2.08^2 + 1.2^2) should be side/2. Son of a gun, using full precision it is.

Each of those triangles is area (2.08 * 1.2)/2, or 1.25 (again, using full precision). There are six of those little triangles for a total area of 7.5.

60 minus 30 minus 7.5 is 22.5! I got the same answer you did! Woo-hoo!

Which means, Mr. PreMath, beware. I am but a beginner but I may someday snatch the pebble from your hand!

Or, maybe not. You've got quite the head start on me.

This was fun. Thank you very much.

[murdock]

Nice! Many thanks, that was great, Sir! 🙂

[APUS_NUNN]

Sehr schöne Lösung! Wer zählen kann, ist gut dran.

[JLvatron]

I did it the long way, but got it right.

[massimogranzotto]

Fortissimo professore, veramente interessante. Grazie

[kunstkritik]

I tried solving it by plugging in the 60 square units into the formular for calculating a regular size hexagon to get the side length and from there I went on calculating the side lengths of the triangle to then calculate its area. However, I ended up at 7.5 square units which is 3 times less your result. So I guess I made a mistake somewhere

[fabrizio]

L'esagono si puo' anche vedere come suddiviso in due trapezi isosceli. Il lato del triangolo verde unisce i due punti medi dei lati obliqui del trapezio; cio' vuol dire che é anche uguale alla semisomma delle basi ovvero 3 l/2 dove l é la misura del lato dell'esagono ricavabile facilmente

[musicsubicandcebu]

Always check the length of the video . . . if short, prob means a simple solution