For a square, find the ratio of the area of its incircle to the area of its circumcircle (answer in the form a:b).

Introduction:
This question tests circle geometry related to a square. A square can have an incircle (a circle inside touching all four sides) and a circumcircle (a circle outside passing through all four vertices). The incircle radius depends on the side length, while the circumcircle radius depends on the diagonal of the square. Since circle area is proportional to the square of its radius, the ratio of areas depends on the ratio of the radii squared. The key is to express both radii in terms of the square’s side length s.

Given Data / Assumptions:

• Let side of square = s• Incircle touches sides, so diameter of incircle = s• Circumcircle passes through vertices, so radius depends on diagonal

Concept / Approach:
Incircle radius r_in = s/2. Square diagonal = s*sqrt(2). Circumcircle radius r_out is half the diagonal, so r_out = (s*sqrt(2))/2 = s/sqrt(2). Then areas are A_in = pi*(r_in^2) and A_out = pi*(r_out^2). Take the ratio and simplify; pi cancels automatically.

Step-by-Step Solution:
Step 1: Compute incircle radius.r_in = s/2Step 2: Compute circumcircle radius using diagonal.Diagonal = s*sqrt(2)r_out = (diagonal)/2 = (s*sqrt(2))/2 = s/sqrt(2)Step 3: Write the areas.A_in = pi*(s/2)^2 = pi*(s^2/4)A_out = pi*(s/sqrt(2))^2 = pi*(s^2/2)Step 4: Take the ratio A_in : A_out.A_in : A_out = (pi*s^2/4) : (pi*s^2/2)Cancel pi*s^2: = (1/4) : (1/2)= 1 : 2

Verification / Alternative check:
Take s=2. Then r_in=1, area_in=pi. Diagonal=2*1.414=2.828, r_out=1.414, area_out=pi*(2)=2pi. Ratio pi:2pi = 1:2. The numeric check matches, confirming correctness.

Why Other Options Are Wrong:
1:1 would require both circles to have equal radius, which is impossible for a square.1:3 and 1:4 exaggerate the difference; the correct circumcircle area is only twice the incircle area.2:3 is not aligned with the exact squared radius relationship.

Common Pitfalls:
• Using the side length instead of diagonal to find circumcircle radius.• Forgetting that circle area depends on radius squared.• Not cancelling common factors like pi and s^2, which simplifies the ratio.

Final Answer:
The ratio of areas (incircle : circumcircle) is 1 : 2.