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The sum of the areas of the circumcircle and incircle of an equilateral triangle is 770 cm^2. Using standard formulas for radii of an equilateral triangle, find the area of the triangle in square centimetres.

Difficulty: Medium

Correct Answer: 147√3

Explanation:


Introduction / Context:

This geometry problem combines properties of an equilateral triangle with the formulas for the radii of its circumcircle and incircle. It tests your ability to connect different formulas, convert between side length and radii, and finally compute the area of the triangle. Such multi-step questions are common in higher level aptitude and entrance tests.



Given Data / Assumptions:

  • The triangle is equilateral, so all sides are equal.
  • The sum of the areas of the circumcircle and incircle is 770 cm^2.
  • We assume π = 22/7 unless stated otherwise.
  • We must find the area of the equilateral triangle in square centimetres.


Concept / Approach:

For an equilateral triangle with side length a, the inradius r and circumradius R have standard formulas: r = (a√3) / 6 and R = (a√3) / 3. The area of a circle is πR^2 or πr^2. Therefore, the sum of the areas of the circumcircle and incircle can be written in terms of a, which then allows us to solve for a^2. Once we know a^2, we can use the equilateral triangle area formula Area = (√3 / 4)a^2.



Step-by-Step Solution:

Step 1: For side length a, the circumradius R = (a√3) / 3 and the inradius r = (a√3) / 6.Step 2: Compute R^2 = (a^2 * 3) / 9 = a^2 / 3 and r^2 = (a^2 * 3) / 36 = a^2 / 12.Step 3: Area of circumcircle = πR^2 = π(a^2 / 3).Step 4: Area of incircle = πr^2 = π(a^2 / 12).Step 5: Sum of areas = π(a^2 / 3 + a^2 / 12) = π a^2 (4/12 + 1/12) = (5π a^2) / 12.Step 6: This sum is given as 770, so set (5π a^2) / 12 = 770.Step 7: Use π = 22/7. Substitute to get (5 * 22 * a^2) / (7 * 12) = 770.Step 8: Simplify numerator: 5 * 22 = 110, so 110a^2 / 84 = 770. Reduce 110 / 84 to 55 / 42.Step 9: So (55 a^2) / 42 = 770. Multiply both sides by 42: 55 a^2 = 770 * 42.Step 10: Compute 770 * 42 = 770 * (40 + 2) = 30800 + 1540 = 32340.Step 11: Therefore a^2 = 32340 / 55 = 588.Step 12: Area of an equilateral triangle is (√3 / 4)a^2 = (√3 / 4) * 588 = 147√3.


Verification / Alternative check:

To check, plug a^2 = 588 back into the sum of areas formula. Sum = (5π a^2) / 12 = (5 * 22/7 * 588) / 12. First compute 588 / 7 = 84, so the expression becomes (5 * 22 * 84) / 12. Then 22 * 84 = 1848, and 5 * 1848 = 9240. Finally, 9240 / 12 = 770, which matches the given value, confirming that a^2 = 588 and the area 147√3 are correct.



Why Other Options Are Wrong:

The other options 125√3, 156√3, 169√3, and 196√3 correspond to different, incorrect values of a^2. If you substitute those values back into the circular area formulas, the sum will not equal 770 cm^2. Only 147√3 passes the consistency check with the given circular areas.



Common Pitfalls:

Typical errors include confusing inradius and circumradius formulas, mixing up the factor 1/3 and 1/6, or using the area formula of a triangle incorrectly. Some learners also forget to use the given approximation π = 22/7 and instead use a rounded decimal, which leads to slightly different numeric results. Following the exact symbolic steps keeps the calculation clean and accurate.



Final Answer:

The area of the equilateral triangle is 147√3 square centimetres.


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