Question: 

An octagon is inscribed in a circle as shown above. What of the area of the octagon?

1. 13+√2
2. 13+4√2
3. 13+6√2
4. 13+12√2
5. 13+15√2

“An octagon is inscribed in a circle as shown above. What of the area” - is a subject covered in the GMAT quantitative reasoning section. A student needs to be knowledgeable in a wide range of qualitative skills in order to successfully complete GMAT Problem Solving questions. There are 31 questions in the GMAT Quant section overall. Calculative mathematical problems must be solved in the GMAT quant topics' problem-solving section using appropriate mathematical skills.

Solutions and Explanation

Approach Solution : 1

Area of octagon = [Area of square whose side length is 3+2√2] - (Area of 4 isosceles right angle triangle whose leg is √2)

Area of octagon = \([3+2√2]^2-4*[\frac{1}{2}*√2*√1]^2= 9+8+12√2-4 = 13+12√2\)

Correct Answer: (D)

Approach Solution : 2

If you are having trouble calculating the exact area of the octagon shown above, you can simply estimate it by assuming that it has a side length of 2.5 (notice that 2.5 is used because it is the average of 2 and 3). The claim is that even though a regular octagon with side length 2.5 can also be divided into 8 triangles, each with a base of 2.5, it will have a fairly similar area to the one depicted above. Each of these triangles is a little bit bigger than the triangle above with a base of 2, but a little bit smaller than the triangle above with a base of 3. Thus, they "average" out" in the same general area. In other words, the area of the octagon shown above is roughly equivalent to a regular octagon with side lengths of 2.5.

We can now apply the knowledge that an ordinary octagon with side length s has the following area:

Area = 2s^2 + 2s^2√2

As a result, if s = 2.5, the area would be 2(2.5)^2 + 2(2.5)^2√2 = 12.5 + 12.5√2.

We see that the value in the fourth option which is 13+12√2
is close to the value we got, so it is the right option.

Correct Answer: (D)

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