Star And Pentagon

Geometry Level 3

Consider a regular pentagon with a regular pentagram in it as shown above.

Which area is larger?

Green Area Blue Area They are equal

2 solutions

Chung Kevin
Oct 23, 2016

If we looks at just 1/5th of the image, and consider the blue isosceles triangle and the green isosceles triangle.

We can calculate that the base angles are $36^\circ$ for the green isosceles triangle and $54^\circ$ for the blue isosceles triangle. Hence, the ratio of their heights is $\sin 36^\circ : \sin 54^ \circ \approx 0.59 : 0.81$ .

As such, the green area is larger than the blue area.

I'm wondering if there is a better way to show this, without resorting to calculating these trigonometric values.

I do not see the blue isosceles triangle you mention. Can you please give a Fig. ? Thanks. My solution is below. There are two isosceles triangles. One the green one and the other green plus blue. Both have the same base. The ratio of their height is Sin36:sin54=59:81.
So the green height is more than half the composite height.
So green area is greater.

Niranjan Khanderia - 4 years, 7 months ago

Ah, I was just connecting the vertices to the center. I believe we have the same approach.

I do not like having to find "sin 36 : sin 54" in this approach, and am wondering if there is another way of visualizing this problem.

Chung Kevin - 4 years, 7 months ago

You could just use similarity without using those trigonometric values.

Chaebum Sheen - 4 years, 7 months ago

Can you elaborate?

Chung Kevin - 4 years, 7 months ago

Note that there are 5 green triangles, and you can carve two green triangles out of the blue part. Then you can separate the blue part into 4 equal triangles whose angles are 36 degrees, 72 degrees and 72 degrees. Thus it generally comes down to comparing these 3 green triangles with these 4 blue triangles. However, the green triangle and blue triangle can be added together to form a larger triangle that is similar to the blue triangle. So adding 3 blue triangles to each side, we just have to compare 7 blue triangles and these 3 triangles similar to the blue triangles. If the ratio of the sides is $x$ , then you can easily see $x^2-x-1=0$ . It can be proven that $x^2 >\frac{7}{3}$ . So the green area is larger. I wish I could better explain this, but I can't provide the figures for it.

Chaebum Sheen - 4 years, 7 months ago

@Chaebum Sheen – You can always upload and image from here , see the toolbar, then copy the "markdown text" and paste it over here.

Pi Han Goh - 4 years, 7 months ago

Kai Ott
Nov 18, 2016

Ah sadly I don't have a picture to this but you can tilt one of the green triangles about the side of an adjacent star-too. The green triangle will then fill one star tip and 2/5 of the inner small pentagon in the star. As there are 5 green triangles and 5 star-tops, the green area cover the blue small pentagon by $5\cdot \frac{2}{5} = 2$ times. Hence the total green area exceeds the blue area by once the inner small pentagon.

Star And Pentagon

2 solutions

0 pending reports