Question: A regular octagon (a polygon with 8 sides of identical length and 8 identical interior angles) is constructed. Next, an equilateral triangle (with sides identical in length to those of the octagon) is attached to each side of the octagon, such that each side of the octagon coincides exactly with the side of the triangle. Finally, each triangle is folded over that coincident side onto the octagon, covering part of the latter’s area. Approximately what proportion of the area of the octagon is left uncovered?

(A) 60%
(B) 50%
(C) 40%
(D) 30%
(E) 20%

Correct Answer: D
Solution and Explanation:
Approach Solution 1:

The two legs of the isosceles triangles will be
1/√2 each (Since sides of an isosceles right triangle are in the ratio
1:1: √2

Area of the isosceles right triangle = (1/2)∗1/√2*1/√2= ¼
Side of square= 1/√2+1+1/√2= 1+√2
Area of square= (1+√2)^2= 3+2√2= 5.8
Area of octagon= 5.8−4∗(1/4)= 4.8

Area of the equilateral triangles= (√3/4)∗1^2 (because area of an equilateral triangle with side s= (√3/4)∗s^2
Area of 8 equilateral triangles= 8∗(√3/4)= 2∗√3= 3.464 
3.6 is 75% of 4.8 so 3.464 will be a little less than 75% of 4.8.

So the leftover area will be a little more than 25%.

Approach Solution 2:

Let's assume that the octagon's each side length is 1
Figure out this octagon’s area, by splitting up the shape into rectangles and right triangles:

Area = Central square + 4 side rectangles + 4 right triangles (45-45-90)
Area = 1 + 4(√2/2) + 4(½)(√2/2)2 = 2 + 2√2 ≈ 2 + 2(1.4) = 4.8 (we can round because the answer is approximate).
Next, figure out the area of the 8 triangles that will be attached and folded over to cover part of the octagon. (By the way, you can see that the triangles won’t touch each other, because the interior angles of the octagon are 135°, and the two triangles only cover 120° when you fold them in.)

Area of 8 equilateral triangles of side length 1 = 8(s2√3/4) = 2√3 ≈ 2(1.7) = 3.4
The uncovered area equals 4.8 – 3.4 = 1.4, and as a percent of 4.8, that area represents
1.4/4.8 = 14/48 = 7/24 ≈ 7/25 = 28/100 = 28%. The closest answer choice is 30%.

Approach Solution 3:

If the side length of the regular octagon is s, then the area of the octagon is given by 2(1 + √2)(s^2). Partitioning the regular octagon into a bunch of isosceles, right triangles, rectangles and a square is one way of obtaining this formula.

Since the regular octagon has 8 sides, 8 equilateral triangles with a side length of s will be drawn and folded over the octagon. Since the area of each equilateral triangle is [(s^2)√3]/4 and since there are 8 such equilateral triangles, the total area of the equilateral triangles is 8 * [(s^2)√3]/4 = 2√3(s^2). Hence, the uncovered area equals 2(1 + √2)(s^2) - 2√3(s^2) = 2(s^2)(1 + √2 - √3). It follows that the ratio we need to approximate is [2(s^2)(1 + √2 - √3)]/[2(1 + √2)(s^2)] = (1 + √2 - √3)/(1 + √2). Since √2 is approximately 1.4 and √3 is approximately 1.7, we get:

(1 + √2 - √3)/(1 + √2)
(1 + 1.4 - 1.7)/(1 + 1.4)
0.7/2.4 ≈ 0.29

Expressed as a percentage, this is 29%, and the closest answer choice is D.

“A regular octagon (a polygon with 8 sides of identical”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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