Hemisphere Geometry Overview

A solid hemisphere is exactly half of a solid sphere. It consists of a curved surface and a flat circular base. The key dimension defining a hemisphere is its radius, denoted by \(r\), which is the distance from the center of the circular base to any point on its circumference, and also from the center to any point on the curved surface.

Volume of a Solid Hemisphere

The volume of a three-dimensional object measures the amount of space it occupies. Since a solid hemisphere is half of a solid sphere, its volume is half the volume of a full sphere. The formula for the volume of a sphere is \(\frac{4}{3}\pi r^3\).

• Volume of a solid hemisphere (\(V\)):
• \(V = \frac{1}{2} \times (\text{Volume of a sphere})\)
• \(V = \frac{1}{2} \times \left(\frac{4}{3}\pi r^3\right)\)
• \(V = \frac{2}{3}\pi r^3\)

Surface Area of a Solid Hemisphere

The surface area of a solid hemisphere refers to the total area of all its surfaces. For a solid hemisphere, this includes two parts:

• The curved surface area (CSA)
• The area of its flat circular base

Let's break down each component:

• Curved Surface Area (CSA) of a hemisphere: This is half the surface area of a full sphere. The surface area of a sphere is \(4\pi r^2\).
• CSA = \(\frac{1}{2} \times (4\pi r^2) = 2\pi r^2\)
• Area of the circular base: The base is a circle with radius \(r\).
• Area of base = \(\pi r^2\)
• Total Surface Area (TSA) of a solid hemisphere: The sum of the curved surface area and the base area.
• TSA = CSA + Area of base
• TSA = \(2\pi r^2 + \pi r^2\)
• TSA = \(3\pi r^2\)

Ratio Calculation: Volume to Surface Area

The question asks for the ratio of volume to surface area of the solid hemisphere. We will use the formulas derived above to calculate this ratio.

• Let the ratio be \(R\).
• \(R = \frac{\text{Volume of solid hemisphere}}{\text{Total Surface Area of solid hemisphere}}\)
• Substitute the formulas:
• \(R = \frac{\frac{2}{3}\pi r^3}{3\pi r^2}\)

Now, let's simplify the expression:

• To simplify the fraction, multiply the numerator by the reciprocal of the denominator:
• \(R = \frac{2}{3}\pi r^3 \times \frac{1}{3\pi r^2}\)
• \(R = \frac{2\pi r^3}{3 \times 3\pi r^2}\)
• \(R = \frac{2\pi r^3}{9\pi r^2}\)
• Cancel out the common terms (\(\pi\) and \(r^2\)):
• \(R = \frac{2r}{9}\)

Relating Ratio to Radius

From our calculation, the ratio of volume to surface area of a solid hemisphere is \(\frac{2r}{9}\). This means the ratio is directly proportional to the radius \(r\). The constant of proportionality, or the factor through which it is related to the radius, is \(\frac{2}{9}\).

Therefore, the relationship is \( \text{Ratio} = \left(\frac{2}{9}\right) \times \text{radius} \).

Summary of Hemisphere Formulas

Here is a quick summary of the formulas used for a solid hemisphere with radius \(r\):

The numerical factor that relates the ratio of volume to surface area to its radius is \(\frac{2}{9}\).