If the diameter of a circle equals the perimeter of an equilateral triangle with side 12 cm, find the ratio of the area of the circle to the area of the triangle.

hard 3 marks Class 5 · Mathematics · Area and its Boundary

Answer

3\pi\sqrt{3}

The correct answer is 3\pi\sqrt{3} — Thus the ratio of the circle’s area to the triangle’s area is $3 π 3 : 1$ .

Step-by-step solution

Compute the perimeter of the triangle: $P = 3 \times 12 = 36$ cm. The diameter of the circle is $36$ cm, so the radius $r = \frac{36}{2} = 18$ cm.
Find area of the circle: $A_{c} = π r^{2} = π \times 1 8^{2} = 324 π$ cm $^{2}$ .
Find area of the equilateral triangle: $A_{t} = \frac{3}{4} a^{2} = \frac{3}{4} \times 1 2^{2} = 363$ cm $^{2}$ .
Take the ratio $\frac{A _{c}}{A _{t}} = \frac{324 π}{36 3} = 9 \frac{π}{3} = 3 π 3$ .
Thus the ratio of the circle’s area to the triangle’s area is $3 π 3 : 1$ .

Common mistake: Using the wrong value of

π

or forgetting the factor

\frac{3}{4}

for the triangle

Concept tested: Ratio of circle to triangle area