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The volume of a hemisphere is calculated using the formula \( V = \frac{2}{3} \pi r^3 \). A hemisphere is essentially half of a sphere. This formula tells us how much space is inside the hemisphere. ### Understanding the Formula- **\( \pi \):** This symbol represents Pi, approximately 3.14159, crucial for calculations involving circles and spheres.- **\( r^3 \):** This is the radius cubed, meaning the radius is multiplied by itself twice more. - **Fraction \( \frac{2}{3} \):** Indicates that the volume of the hemisphere is two-thirds the volume of a full sphere due to its half nature.Using this formula helps compare the space inside a hemisphere versus other shapes with the same radius. It gives us a clear numerical value that can be compared with other volumes, such as that of a cone.

The volume of a cone can be calculated using the formula \( V = \frac{1}{3} \pi r^2 h \). Cones have a circular base and converge to a single point, and this formula expresses the amount of space inside them.### Key Elements in the Formula- **\( \pi \):** As with other shapes involving circles, Pi is essential for calculations.- **\( r^2 \):** This refers to the radius squared or the radius multiplied by itself. This part measures the base's area.- **\( h \):** Represents the height, or the distance from the base to the tip of the cone.- **Fraction \( \frac{1}{3} \):** Indicates that a cone's volume is one-third of the volume of a cylinder with the same base and height.This formula is helpful when determining how much a cone holds or for comparisons with other shapes, like a hemisphere, especially when similar size conditions are applied.

In the scenario where the radius \( r \) equals the height \( h \) for both the hemisphere and the cone, the relationship significantly impacts their volume comparison.### Setting \( r = h \)- In the cone formula \( h \) is replaced by \( r \), resulting in \( V = \frac{1}{3} \pi r^3 \).- The hemisphere's volume remains \( V = \frac{2}{3} \pi r^3 \).### Why the Hemisphere's Volume is Larger- With equal radius and height, the hemisphere's formula essentially doubles the coefficient of the cone's formula, since two-thirds is greater than one-third.- Resultantly, for any given positive value of \( r \), the hemisphere always has a larger volume than the cone under these conditions.This comparison demonstrates how the radius and height conditions play a critical role in determining the volumes of these two solids.