Question

a sphere is cut into 8 congruent pieces. the radius of the sphere is 8 centimeters. one piece is shown in the diagram. what is the surface area of one piece of the sphere, as shown in the diagram? express the answer in terms of π. 64π cm² 80π cm² 256π cm² 384π cm²

Explanation:

Step1: Calculate sphere surface area

The formula for the surface area of a sphere is $SA_{sphere}=4\pi r^2$. Substitute $r=8$ cm:
$$SA_{sphere}=4\pi (8)^2=4\pi \times 64=256\pi \text{ cm}^2$$

Step2: Find curved area of one piece

Since the sphere is split into 8 congruent pieces, the curved surface area of one piece is $\frac{1}{8}$ of the total sphere surface area:
$$SA_{curved}=\frac{256\pi}{8}=32\pi \text{ cm}^2$$

Step3: Calculate flat face area

Each piece has 3 congruent right triangular flat faces. Area of one triangle is $\frac{1}{2} \times 8 \times 8=32$ cm². Total flat area:
$$SA_{flat}=3 \times 32=96 \text{ cm}^2$$

Step4: Convert flat area to $\pi$ equivalent

Wait, correction: The flat faces are **quarter-circles? No, no—each flat face is a right isosceles triangle? No, actually, when cutting a sphere into 8 congruent octants, each flat face is a semicircle? No, correct: Each octant has 3 flat faces, each is a quarter of a circle with radius 8? No, no—each flat face is a right triangle? No, no, the flat faces are circular segments? No, correct approach: When you cut a sphere into 8 congruent pieces (octants), each piece has 3 flat faces, each is a quarter-circle? No, no, the flat face is a semicircle? Wait no, the area of each flat face is a quarter of a circle with radius r: $\frac{1}{4}\pi r^2$. So 3 flat faces:
$$SA_{flat}=3 \times \frac{1}{4}\pi (8)^2=3 \times \frac{64\pi}{4}=3 \times 16\pi=48\pi \text{ cm}^2$$

Step5: Total surface area of one piece

Add curved and flat surface areas:
$$SA_{total}=32\pi + 48\pi=80\pi \text{ cm}^2$$

Answer:

$80\pi$ cm²