A Regular Hexagon has a Perimeter of 30 units GMAT Problem Solving

Question: A regular hexagon has a perimeter of 30 units. What is the sum of the lengths of all its diagonals?

1. 30 + \(\sqrt3\)
2. 30 (1 + \(\sqrt3\))
3. 30 (1 +\(\sqrt2\))
4. 20 \(\sqrt3\)
5. 30

‘A regular hexagon has a perimeter of 30 units.' - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

This question can be solved by only one approach.

Given to us that a hexagon has a perimeter of 30 units. It has asked to find out the sum of the lengths of the diagonals.
This is a question from geometry.
A hexagon is a figure having six sides.
The perimeter is given to be 30 units
Let the side of the hexagon be x.
Perimeter of hexagon = 6x = 30
We get,
X = 5
The side of the hexagon is 5.
A hexagon has 9 diagonals.

Out of 9 diagonals, 3 are of one kind and the other 6 are of another kind.
Each of the three red diagonals is of length = 2 * side = 2 * 5 = 10 (since a regular hexagon is made of 6 equilateral triangles)

Each of the three blue diagonals will be equal to 2*(\(\sqrt3/2 * side\)) = 2(\(\sqrt3/2*5\)) = \(5\sqrt3\)

It should be noted that in a triangle having sides 30, 90, and 60 degrees the sides are divided in the ratio 1:2:\(\sqrt3\) .

Half of the blue diagonal is the leg opposite to 60 degrees. So it is equal to side *\(\sqrt3\)/2.

Now the sum of length of all the diagonal will be -

3 * 10 + 6 * \(5\sqrt3\) = 30 + \(30\sqrt3\) = 30 (1+\( \sqrt3\))

Therefore option B is the correct answer.

Correct Answer: B

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