Sphere Surface Area Math Example 5

Follow the full solution, then compare it with the other examples linked below.

Example 5

hard

Two spheres have radii in the ratio

2:3

. Find the ratio of their surface areas. If the smaller sphere has a surface area of

64\pi

cm², find the surface area of the larger sphere.

Solution

1
Step 1: Surface area scales with the square of the radius. If the radii are in ratio $2:3$ , the surface areas are in ratio $2^2:3^2 = 4:9$ .
2
Step 2: Set up the proportion: $\frac{SA_{\text{small}}}{SA_{\text{large}}} = \frac{4}{9}$ . Substitute $SA_{\text{small}} = 64\pi$ : $\frac{64\pi}{SA_{\text{large}}} = \frac{4}{9}$ .
3
Step 3: Cross-multiply: $SA_{\text{large}} = \frac{9 \times 64\pi}{4} = 9 \times 16\pi = 144\pi$ cm².
4
Step 4: Verify: the smaller sphere has $SA = 4\pi r^2 = 64\pi$ , so $r^2 = 16$ , $r = 4$ . The larger has $r = 6$ (ratio 4:6 = 2:3 ✓), and $SA = 4\pi(36) = 144\pi$ ✓.

Answer

Ratio of surface areas is

4:9

; larger sphere has

SA = 144\pi

cm².

Surface area is proportional to the square of the radius. A 2:3 radius ratio gives a 4:9 area ratio. Scaling up the smaller sphere's area of 64π by the factor 9/4 gives 144π cm² for the larger sphere.