How do you find the capacity of a hemispherical cup?

The capacity of a hemispherical cup is determined by its volume, which can be calculated using the formula for half a sphere: V = (2/3)πr³.

Understanding Hemispherical Cup Capacity

The capacity of any container refers to the maximum amount of substance (liquid, granular material, etc.) it can hold. For a hemispherical cup, this is equivalent to finding the volume of a perfect hemisphere. A hemisphere is simply half of a sphere.

To accurately find this capacity, you need just one measurement: the radius of the hemisphere.

The Core Formula

The fundamental formula to calculate the volume (capacity) of a hemispherical cup is:

V = (2/3)πr³

Where:

V represents the volume or capacity of the cup.
π (pi) is a mathematical constant approximately equal to 3.14159.
r is the radius of the hemisphere (the distance from the center to any point on the rim or curved surface).
r³ means r multiplied by itself three times (r * r * r).

The resulting volume will be in cubic units (e.g., cubic centimeters (cm³), cubic inches (in³), cubic meters (m³)).

Practical Steps to Measure and Calculate

Finding the capacity of your hemispherical cup involves a few straightforward steps:

Measure the Diameter:
- Carefully measure the diameter of the cup's opening. This is the widest distance across the circular rim. Use a ruler or calipers for accuracy.
- Example: If the diameter is 10 cm.
Calculate the Radius:
- The radius (r) is half of the diameter.
- Example: r = Diameter / 2 = 10 cm / 2 = 5 cm.
Apply the Formula:
- Substitute the radius value into the formula V = (2/3)πr³.
- Example:
  V = (2/3) * π * (5 cm)³
  V = (2/3) * π * 125 cm³
  V ≈ (2/3) * 3.14159 * 125 cm³
  V ≈ 2.09439 * 125 cm³
  V ≈ 261.799 cm³
Convert to Common Units (Optional):
- Often, capacity is expressed in milliliters (mL) or liters (L).
- Remember that 1 cm³ = 1 mL.
- Example: 261.799 cm³ ≈ 261.8 mL.
- Since 1000 mL = 1 L, 261.8 mL = 0.2618 L.

Example Calculation

Let's calculate the capacity of a hemispherical cup with a measured diameter of 12 cm.

Find the Radius (r):
- Diameter = 12 cm
- Radius (r) = 12 cm / 2 = 6 cm
Apply the Volume Formula:
- V = (2/3)πr³
- V = (2/3) π (6 cm)³
- V = (2/3) π (6 6 6) cm³
- V = (2/3) π 216 cm³
- V = 2 π (216 / 3) cm³
- V = 2 π 72 cm³
- V = 144π cm³
Approximate the Value:
- Using π ≈ 3.14159:
- V ≈ 144 * 3.14159 cm³
- V ≈ 452.39 cm³
Convert to Milliliters:
- Since 1 cm³ = 1 mL, the capacity is approximately 452.39 mL.

Capacity of Complex Vessels with Hemispherical Portions

While a simple hemispherical cup has a straightforward volume, many containers feature hemispherical elements as part of a more complex design. For instance, calculating the capacity of a vessel that combines a cylindrical portion with a hemispherical portion requires a more specific formula. In such cases, if the vessel has a common radius 'r' and a cylindrical height 'h', its capacity can be determined as πr²/3 (3h - 2r) cubic units. This specific formula accounts for the combined geometric properties of the cylindrical and hemispherical components within a single unit.

Summary of Key Geometric Formulas for Capacity

Understanding these fundamental formulas is crucial for calculating the capacity of various shapes encountered in everyday objects.

Shape	Formula for Volume (Capacity)	Variables
Sphere	V = (4/3)πr³	r = radius
Hemisphere	V = (2/3)πr³	r = radius
Cylinder	V = πr²h	r = radius, h = height
Cone	V = (1/3)πr²h	r = radius, h = height
Rectangular Prism	V = lwh	l = length, w = width, h = height

Calculating the capacity of a hemispherical cup is a straightforward application of geometry, requiring only a simple measurement of its radius.