Case 1
- Bottom square side a
- 6
- Top square side b
- 4
- Vertical height h
- 9
228.000000
Open with these valuesPyramid Frustum Volume Calculator
228.000000units³
Result: 228.000000 units³A frustum is a pyramid with the tip cut off parallel to the base. For square bases the volume is h ÷ 3 × (a² + a×b + b²): with a = 6, b = 4 and h = 9 that is 3 × 76 = 228 cubic units. Enter the two square side lengths, not their areas.
The numbers at a glance
Held fixed: Bottom square side a 6.000, Top square side b 4.000.
| Vertical height h | Result |
|---|---|
| 2.500 | 63.333333 |
| 5.000 | 126.666667 |
| 7.500 | 190.000000 |
| 9.000Your value | 228.000000 |
| 10.000 | 253.333333 |
| 12.500 | 316.666667 |
| 15.000 | 380.000000 |
| 17.500 | 443.333333 |
Worked examples
Case 2
- Bottom square side a
- 10
- Top square side b
- 6
- Vertical height h
- 8
522.666667
Open with these valuesCase 3
- Bottom square side a
- 5
- Top square side b
- 2
- Vertical height h
- 7
91.000000
Open with these valuesHow it's calculated
V = h ÷ 3 × (a² + a×b + b²)
- StepMeasure one side of the larger bottom square.
- StepMeasure one side of the smaller top square.
- StepMeasure straight up from the bottom square to the top one.
- ResultSquare both sides, add their product, multiply by h ÷ 3.
Reference table
| Bottom a, top b, height h | Exact | Slant height | Lateral surface | Total surface | Volume |
|---|---|---|---|---|---|
| 5, 2, 7 | 7 ÷ 3 × 39 | 7.158911 | 100.224747 | 129.224747 | 91 |
| 6, 4, 9 | 9 ÷ 3 × 76 | 9.055385 | 181.107703 | 233.107703 | 228 |
| 8, 8, 5 | 5 ÷ 3 × 192 | 5 | 160 | 288 | 320 |
| 10, 6, 8 | 8 ÷ 3 × 196 | 8.246211 | 263.878760 | 399.878760 | 522.666667 |
| 12, 4, 10 | 10 ÷ 3 × 208 | 10.770330 | 344.650548 | 504.650548 | 693.333333 |
Questions
How do I calculate the volume of a pyramid frustum?
Use V = h ÷ 3 × (a² + a×b + b²), with a the bottom square side, b the top square side and h the height. For a = 6, b = 4, h = 9 that is 3 × (36 + 24 + 16) = 3 × 76 = 228 cubic units.
What is a pyramid frustum?
A frustum is a pyramid whose top has been sliced off by a cut parallel to the base. A square one has a large square bottom, a smaller square top and four sloping trapezoidal faces. Lampshades, planters, cake tiers and hoppers are everyday frustums.
Do I enter the side lengths or the base areas?
The side lengths of the two squares. The general formula uses the areas, h ÷ 3 × (A₁ + A₂ + √(A₁A₂)), but for squares A₁ = a², A₂ = b² and √(A₁A₂) = a×b, which is exactly the form used here.
What is the slant height of a frustum?
It is the height measured down the middle of a sloping face, √(h² + ((a − b) ÷ 2)²), where (a − b) ÷ 2 is how far each side steps inward. For a = 6, b = 4, h = 9 it is √82 ≈ 9.055385, and the table above lists it for every example.
What is the difference between lateral and total surface?
The lateral surface is only the four sloping faces, 2 × (a + b) × slant height. The total adds the bottom square a² and the top square b². Both stand in the table above; use the lateral figure when top and bottom are open.
Which units does the answer use?
Whatever you entered, cubed. All three inputs must share one unit, and the slant height in the table comes back in that same unit while the surfaces are in its square.
Sources and last check
Information, not professional advice.