4.5. Surface Area and Volume of a Sphere and Hemisphere

● Sphere radius = rr. ○ Surface Area = 4πr² ○ Volume = $\frac{4}{3} \pi r^3$

A sphere is a perfectly round three-dimensional shape, like a basketball. To calculate the surface area, we use the formula $4\pi r^2$, where 'r' is the radius of the sphere. This formula helps find the total area that covers the outer surface of the sphere. For the volume, we use the formula $\frac{4}{3} \pi r^3$, which gives us the total space contained within the sphere. Both formulas depend on the radius, indicating how size affects both surface area and volume.

Think of a basketball: The surface area would tell you how much material is needed to cover the outside, while the volume would tell you how much air is inside it. If we know the radius (for example, 6 cm), we can find out how large the basketball is not just outside, but also inside where the air is stored.

● Hemisphere radius = rr. ○ Curved Surface Area = 2πr² ○ Total Surface Area = 3πr² ○ Volume = $\frac{2}{3} \pi r^3$

A hemisphere is essentially half of a sphere, similar to cutting a basketball in half. To find the curved surface area of the hemisphere, we use the formula $2\pi r^2$. The total surface area, which includes the flat circular base, is given by $3\pi r^2$. When it comes to calculating the volume, the formula is $\frac{2}{3} \pi r^3$, which represents the space contained within the hemisphere.

Imagine a half of a dome-shaped glass or the top half of a snow globe. The curved surface area tells us how much glass is covering that half, while the total surface area considers both the curved part and the flat base at the bottom. The volume tells us how much liquid could fit inside if the hemisphere were filled, providing a practical way to understand the amount of space occupied.

✦ Example: Calculate the surface area and volume of a sphere of radius 6 cm. Solution: Surface Area = 4×(22/7)×6² = 4×(22/7)×36 = 1085.71 cm² Volume = (4/3)×(22/7)×6³ = (4/3)×(22/7)×216 = 905.14 cm³

Let's calculate the surface area and volume of a sphere with a radius of 6 cm. For the surface area, we multiply 4 by $\frac{22}{7}$ and then by the square of the radius (6 cm), leading us to 1085.71 cm². For the volume, we take $\frac{4}{3}$, multiply it by $\frac{22}{7}$, and then by the cube of the radius (6 cm). This results in a volume of 905.14 cm³. These calculations demonstrate how to apply the formulas to find specific measurements.

If we think of a round balloon inflated to a size with a radius of 6 cm, the surface area tells you how much surface there is to decorate. Meanwhile, the volume indicates the amount of air inside. Thus, knowing these measurements helps with both crafting and understanding the balloon's capacity.

✦ Example: Calculate the surface area and volume of a hemisphere with radius 6 cm. Solution: Curved Surface Area = 2×(22/7)×6² = 2×(22/7)×36 = 152.57 cm² Total Surface Area = 3×(22/7)×6² = 3×(22/7)×36 = 226.79 cm² Volume = (2/3)×(22/7)×6³ = (2/3)×(22/7)×216 = 272.73 cm³

Now, we will calculate the surface area and volume of a hemisphere with a radius of 6 cm. First, to get the curved surface area, we calculate $2×(\frac{22}{7})×6²$, which gives us approximately 226.79 cm². The total surface area is calculated by adding the curved surface area and the base area, leading to approximately 226.79 cm². Finally, the volume of the hemisphere is determined using the formula $\frac{2}{3}×(\frac{22}{7})×6³$, which will yield around 272.73 cm³. Each of these measurements provides useful information about the hemisphere's dimensions.

Imagine a half-sphere container perfect for holding ice cream. The curved surface area tells us how much area there is to enjoy the beautiful design, while the total surface area includes everything it sits on. The volume informs us about how much ice cream can fill it up. So, when you get a scoop from that container, you can visualize all those measurements coming together in a delicious way!

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