The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having area equal to the sum of the areas of the two circles.

Let r1 and A1 be the radius and area, respectively of the first circle and r2 and A2 be the radius and area, respectively of the second circle. Let r3 and A3 be the radius and area, respectively of the required circle.

For the first circle, r1 = 8 cm (Given)
A1 = πr1² ∴ A1 = π × 8² cm² = 64π cm² .. (1)

For the second circle, r2 = 6 cm (Given)
A2 = πr2² ∴ A2 = π × 6² cm² = 36π cm² .. (2)

Now, for the required circle, A3 = A1 + A2
Given ⇒ A3 = 64π + 36π
From (1) and (2) ⇒ A3 = 100π cm²
∴ πr3² = 100π
⇒ r3² = 100
⇒ r3 = 10

The radius of the required circle is 10 cm.