In a right-angled triangle, the two sides forming the right angle are 3 cm and 4 cm long. What is the area of its circumcircle in square centimetres?

Introduction / Context:
This geometry question combines properties of right angled triangles with the concept of a circumcircle. You are given the two perpendicular sides of the triangle and are asked to find the area of its circumcircle. It tests knowledge of the relationship between the hypotenuse and the circumradius in a right triangle, as well as correct use of the area formula for a circle.

Given Data / Assumptions:

- The triangle is right angled.
- The legs forming the right angle measure 3 cm and 4 cm.
- The circumcircle is the circle passing through all three vertices of the triangle.
- We must find the area of this circumcircle in square centimetres.
- π is treated as the standard constant in the area formula for a circle.

Concept / Approach:
First, we find the hypotenuse of the right triangle using the Pythagoras theorem. A key property of a right triangle is that the circumcentre lies at the midpoint of the hypotenuse, and the circumradius equals half the length of the hypotenuse. After obtaining the circumradius r, the area of the circumcircle is A = π * r^2. Substituting the correct radius gives the required area.

Step-by-Step Solution:
Step 1: Let the legs be a = 3 cm and b = 4 cm.Step 2: Hypotenuse c is found using Pythagoras theorem: c^2 = a^2 + b^2.Step 3: Compute c^2 = 3^2 + 4^2 = 9 + 16 = 25, so c = 5 cm.Step 4: For a right triangle, circumradius r = c / 2 = 5 / 2 = 2.5 cm.Step 5: Area of circumcircle A = π * r^2.Step 6: Compute r^2 = 2.5^2 = 6.25.Step 7: Therefore, A = 6.25π sq. cm.

Verification / Alternative check:
Another way to confirm the radius is to recall that in a right triangle the circle drawn with diameter as the hypotenuse passes through all three vertices. This is precisely the circumcircle. Since the hypotenuse is 5 cm, the diameter of the circumcircle is 5 cm and the radius is half, which again gives r = 2.5 cm. Substituting r into the area formula π * r^2 reproduces 6.25π sq. cm.

Why Other Options Are Wrong:
5π sq. cm would correspond to a radius whose square is 5, not 6.25. Values like 7.5π sq. cm, 10π sq. cm and 12.5π sq. cm are based on incorrect radii, usually arising from mixing up diameter and radius or skipping the division by 2. None of these match the correct r derived from the Pythagoras theorem and the right triangle circumcircle property.

Common Pitfalls:
Common mistakes include taking the hypotenuse itself as the radius instead of half the hypotenuse, or using the wrong formula for area (such as π * d^2 instead of π * r^2). Some learners also miscalculate 2.5^2, writing it as 5 instead of 6.25. Keeping track of diameter versus radius and carefully squaring the radius prevents such errors.

Final Answer:
Hence, the area of the circumcircle is 6.25π sq. cm.