Geometry Help

[apcmlc]

I'm trying to help my son with a math problem and not sure if I am on the right track to explain how to do it.

A 6 pointed regular star consists of 2 interlocking equilateral triangles. What is the ratio of the area of the entire star to the area of one of the equilateral triangles?

I think you need to find the area of one of the equilateral triangles first. Then you need to find the area of the hexagon that is created when the 2 triangles interlock. Then you can double the area of the triangles and subtract out the area of the hexagon to find the area of the star. Then you can create your ratio? Is this right?

 

[galactus]

Yes, you could find the area of the inner hexagon, then the area of the 6 'points'

Add the two together to find the area of the entire star.

Then, find the area of one equilateral and divide into the above result.

A handy formula for the area of any regular polygon given a side length is

\(\displaystyle Area=\frac{s^{2}\cdot n}{4tan(\frac{\pi}{n})}\)

where n is the number of sides and s is the length of a side. In this case, n=6

For this problem, try using a side length of s=1. It may make the problem a little easier.

Also, the area of an equilateral triangle is given by \(\displaystyle \frac{\sqrt{3}}{4}\cdot s^{2}\), where s is a side length.

 

[Deleted member 4993]

I'm trying to help my son with a math problem and not sure if I am on the right track to explain how to do it.

A 6 pointed regular star consists of 2 interlocking equilateral triangles. What is the ratio of the area of the entire star to the area of one of the equilateral triangles?

I think you need to find the area of one of the equilateral triangles first. Then you need to find the area of the hexagon that is created when the 2 triangles interlock. Then you can double the area of the triangles and subtract out the area of the hexagon to find the area of the star. Then you can create your ratio? Is this right?

Yes...

If I understand correctly the meaning of regular polygon (being a figure with six-sided symmetry) then the length of each side of the hexagon is 1/3 that of the base of the equilateral triangles.

 

[soroban]

Hello, apcmlc!

We can "eyevall" the problem . . .

A 6-pointed regular star consists of 2 interlocking equilateral triangles.
What is the ratio of the area of the entire star to the area of one of the equilateral triangles?

. . \(\displaystyle \begin{array}{c} \wedge \\[-3mm] -\!\!\!-\!\!\!- \\[-3mm] \vee \!\! \vee \!\! \vee \\[-3mm] -\!- \\[-3mm] \wedge \!\! \wedge \!\! \wedge \\[-3mm] -\!\!\!-\!\!\!- \\[-3mm] \vee \end{array}\)

The star is composed of 12 equilateral triangles.
An equilateral triangle is composed of 9 equilateral triangles.

\(\displaystyle \text{Therefore: }\;\;{\text{Area of star} : \text{Area of triangle} \;=\;12: 9 \;=\; 4 : 3\)

 

[galactus]

Soroban. A clever and simplistic way to go about it.

I think I prefer your method over my Rube Goldbergian menagerie

 

[Deleted member 4993]

I like that method too .... I am an engineer .... I was supposed to see that.....