A triangle has sides of lengths 6 cm, 8 cm and 10 cm and is inscribed in a circle. What is the area, in square centimetres, of the circumcircle of this triangle?

Introduction / Context:
In this question we relate a right angled triangle to its circumcircle. When a triangle is right angled, the circumcircle has a very special property: the hypotenuse is the diameter of the circumcircle. This allows us to find the radius quickly and then compute the area of the circle. Problems of this type are common in aptitude tests and help reinforce the connection between triangle geometry and circles.

Given Data / Assumptions:

• The sides of the triangle are 6 cm, 8 cm and 10 cm.
• The triangle is inscribed in a circle, so the circle is its circumcircle.
• We must find the area of this circumcircle in square centimetres.
• No value of π is explicitly given, so we can safely use π = 22/7 to match the answer forms expressed as a rational multiple of 1/7.

Concept / Approach:
First, notice that 6, 8 and 10 form a Pythagorean triple: 6^2 + 8^2 = 36 + 64 = 100 = 10^2. This means the triangle is right angled, with the side of length 10 cm as the hypotenuse. For a right angled triangle, the circumcircle has a diameter equal to the hypotenuse. Thus, the radius r of the circumcircle is half the hypotenuse. Once the radius is known, the area of the circle can be found using the formula: Area = π * r^2

Step-by-Step Solution:
Step 1: Verify the triangle is right angled. Compute 6^2 + 8^2 = 36 + 64 = 100 and 10^2 = 100, so the triangle is right angled with hypotenuse 10 cm. Step 2: Use the property of the circumcircle of a right angled triangle: the hypotenuse is the diameter. Thus, diameter = 10 cm, so radius r = 10 / 2 = 5 cm. Step 3: Write the area formula for the circle: Area = π * r^2. Substitute r = 5 cm: Area = π * 5^2 = 25π. Step 4: Express area as a rational number using π = 22/7 to match the form of the options. Area = 25 * (22 / 7) = (25 * 22) / 7 = 550 / 7 square centimetres. Step 5: Compare with the options and choose 550/7 sq cm.

Verification / Alternative check:
We can quickly approximate 550/7. Since 7 * 78 = 546 and 7 * 79 = 553, 550/7 is a little more than 78.5 square centimetres. If we approximate π as 3.14, the area from 25π is 25 * 3.14 = 78.5 square centimetres, which matches the approximation from 550/7. This confirms that the conversion from 25π to 550/7 is correct and consistent with the approximate decimal value.

Why Other Options Are Wrong:
Option 1: 275/7 sq cm represents half of the correct area and would correspond to 12.5π, which is not the area of a circle with radius 5 cm. Option 3: 2200/7 sq cm is four times the correct answer and would correspond to 100π, as if the radius were 10 cm instead of 5 cm. Option 4: 1100/7 sq cm is twice the correct answer and would correspond to 50π, as if the radius were √50, which is not implied by the given triangle. Option 5: None of these is incorrect because 550/7 sq cm is among the options and matches the correct calculation.

Common Pitfalls:
A frequent mistake is failing to recognize that the triangle is right angled. Without this insight, some students attempt to apply a general circumradius formula, which is more time consuming and error prone. Another pitfall is to treat 10 cm as the radius instead of the diameter, which leads to an area four times too large. Students might also forget to square the radius when applying the area formula or mishandle the fraction when converting 25π into a rational form using π = 22/7. Staying organized with each step and recalling special properties of right angled triangles helps avoid these errors.

Final Answer:
The area of the circumcircle is 550/7 sq cm.