The corners of a square of side a are cut away equally so that the remaining figure is a regular octagon. What is the length of each side of the regular octagon, in terms of a?

Introduction / Context:
This problem is a classic geometry construction question. A regular octagon is formed by cutting congruent right isosceles triangles from the corners of a square. The goal is to relate the side length of the original square to the side length of the resulting regular octagon. It tests understanding of symmetry and basic coordinate or segment geometry.

Given Data / Assumptions:

• We have a square with side length a.
• At each corner of the square, a congruent isosceles right triangle is cut away.
• The remaining figure is a regular octagon, so all eight sides of the octagon are equal.
• The cuts along each side of the square are of equal length from the corners.
• We need the side length of the regular octagon in terms of a.

Concept / Approach:
Let x be the length along each side of the square that is cut off from each corner. After cutting, each side of the octagon arises in two ways:

• One type is the middle segment on each side of the original square, which has length a − 2x.
• The other type is the hypotenuse of each cut off right isosceles triangle, which has length x√2.

For a regular octagon, all sides must be equal, so these two expressions must match: a − 2x = x√2 Solving this equation for x and then for the common side gives the required formula.

Step-by-Step Solution:
Step 1: Let the square side be a and the length cut from each end of a side be x. Step 2: The central segment on each side of the square after cutting has length a − 2x. Step 3: Each cut off corner is an isosceles right triangle with legs x and x, so its hypotenuse (which becomes a side of the octagon) has length x√2. Step 4: For a regular octagon, these must be equal: a − 2x = x√2. Step 5: Rearrange: a = 2x + x√2 = x(2 + √2), so x = a / (2 + √2). Step 6: The side length s of the octagon is x√2. Step 7: Substitute x: s = (a / (2 + √2)) * √2. Step 8: Simplify by rationalizing: multiply numerator and denominator by (2 − √2). Step 9: s = a√2(2 − √2) / (4 − 2) = a√2(2 − √2) / 2. Step 10: Expand numerator: √2 * 2 = 2√2 and √2 * √2 = 2, so we get (2√2 − 2) / 2 = √2 − 1. Step 11: Therefore, s = a(√2 − 1).

Verification / Alternative check:
We can test with a convenient numerical value, for example a = 1. Then s = (√2 − 1). Computing x from a = x(2 + √2), we get x = 1 / (2 + √2). The central segment a − 2x and hypotenuse x√2 both evaluate numerically to the same value (√2 − 1), which confirms the derivation.

Why Other Options Are Wrong:
a(√3 − 1): This does not arise from any geometric relation in this construction and is numerically too large.
a/(√2 + 2): This is actually the length x of the cut segments along the square sides, not the octagon side length.
a/3 and a/2: These are simple fractions of the square side with no connection to the angle of 45 degrees in the corner triangles and do not satisfy the equality a − 2x = x√2 with x chosen accordingly.

Common Pitfalls:
Students often confuse x (the cut length) with the octagon side, or forget that the corner triangles are right isosceles, leading to incorrect hypo formulas. Another error is failing to enforce that the octagon has all sides equal, which is the defining property. Carefully labeling segments and writing the equality condition avoids these mistakes.

Final Answer:
The side length of the regular octagon is a(√2 − 1).