Calculating the volume of a shape
Units:
Note that units are shown for convenience but do not affect the calculations. The units are in place to give an indication of the order of the results such as ft, ft2 or ft3. For example, if you are starting with mm and you know a and h in mm, your calculations will result with V in mm3.
Below are the standard formulas for volume.
Volume Formulas:
Capsule Volume
- Volume = πr2((4/3)r + a)
- Surface Area = 2πr(2r + a)
Circular Cone Volume & Surface Area
- Volume = (1/3)πr2h
- Lateral Surface Area = πrs = πr√(r2 + h2)
- Base Surface Area = πr2
- Total Surface Area = L + B = πrs + πr2 = πr(s + r) = πr(r + √(r2 + h2))
Triangular Prism Volume
\[ V = \frac{1}{2} \times b \times h \times l \]
Example
Here's a volume calculation example for a rectangular prism:
"A rectangular prism has a length of 5, width of 3, and height of 8."
The formula to calculate the volume of a rectangular prism is:
V = l × w × h
In this case, the length (l) is 5, width (w) is 3, and height (h) is 8. Plugging these values into the formula:
V = 5 × 3 × 8 = 120 cubic units
The volume of this rectangular prism is 120 cubic units. This demonstrates how simple multiplication can help us calculate the volume of 3D shapes.
| Shape | Formula | Volume | Dimensions |
|---|---|---|---|
| Cube | V = a³ | Example: 3³ = 27 | Side length (a) |
| Rectangular Prism | V = l × w × h | Example: 4 × 5 × 6 = 120 | Length (l), Width (w), Height (h) |
| Sphere | V = (4/3) × π × r³ | Example: (4/3) × π × 3³ ≈ 113.1 | Radius (r) |
| Cylinder | V = π × r² × h | Example: π × 3² × 7 ≈ 197.9 | Radius (r), Height (h) |
| Triangular Prism | V = (1/2) × b × h × l | Example: (1/2) × 5 × 4 × 10 = 100 | Base (b), Height (h), Length (l) |
