The Frustum Surface Area Calculator calculates the total and lateral surface area of a conical frustum (a cone with its top cut off) or a square pyramid frustum (a truncated pyramid).
Frustum Surface Area Calculator
Surface area of a conical frustum (truncated cone) or a square pyramid frustum (truncated pyramid).
For a conical frustum, the curved side is π(R + r)s and the two circular ends are πR² and πr². You can enter the two radii with either the vertical height or the slant height; the calculator converts between them. Lampshades, buckets, funnels and tapered hoppers are all frustums.
Formula
Conical frustum:
A = π(R + r)s + π(R² + r²), with s = √(h² + (R − r)²)
Square pyramid frustum:
A = 2(a + b)s + a² + b², with s = √(h² + ((a − b) ÷ 2)²)
Where:
- R, r = bottom and top radius
- a, b = bottom and top side of the square ends
- h = vertical height; s = slant height
- Lateral area = the sloping sides only: π(R + r)s or 2(a + b)s
How the Calculation Works
Step 1: Find the slant height from the vertical height and the difference between the radii (or half the difference between the sides).
Step 2: Calculate the lateral area of the sloping sides.
Step 3: Add the areas of the top and bottom ends. Leave out an end if it is open (for a bucket or lampshade, use the lateral area plus only the ends you need).
Variables Explained
| Variable | Meaning | Unit |
|---|---|---|
| R, r | Bottom and top radius | cm, m, in, ft |
| a, b | Bottom and top square side | cm, m, in, ft |
| h | Vertical height | cm, m, in, ft |
| s | Slant height | cm, m, in, ft |
| A | Total surface area | cm², m², in², ft² |
Example Calculation
Conical frustum
Given: R = 6 cm, r = 3 cm, h = 4 cm
Slant height: s = √(4² + 3²) = 5 cm
Lateral area: π × (6 + 3) × 5 ≈ 141.37 cm²
Ends: π × (36 + 9) ≈ 141.37 cm²
Result: A ≈ 282.74 cm²
Square pyramid frustum
Given: bottom side = 8 cm, top side = 4 cm, h = 5 cm
Slant height: s = √(5² + 2²) = √29 ≈ 5.385 cm
Calculation: A = 2 × (8 + 4) × 5.385 + 64 + 16 = 129.24 + 80
Result: A ≈ 209.24 cm²
Common Area Conversion Table
| Unit | Equivalent |
|---|---|
| 1 m² | 10,000 cm² |
| 1 ft² | 144 in² |
| 1 m² | 10.7639 ft² |
Shape Properties
A frustum is what remains when a cone or pyramid is cut by a plane parallel to its base. It has two parallel ends of different sizes. When the top radius shrinks to zero, the frustum becomes a full cone; when both radii are equal, it becomes a cylinder.
Dimensions Explained
Diameter vs radius
If you measured the top and bottom diameters, halve both before entering them.
Vertical height vs slant height
The vertical height is measured straight between the ends; the slant height is measured along the side. Use whichever you have — the calculator has a mode for each.
Frequently Asked Questions
How do you find the surface area of a frustum?
Add the curved area π(R + r)s to the two end circles πR² + πr².
What is the lateral surface area of a frustum of a cone?
L = π(R + r)s, where s is the slant height.
How do you find the slant height of a frustum?
s = √(h² + (R − r)²), using the vertical height and the difference between the radii.
What is the surface area of a truncated pyramid?
For square ends: A = 2(a + b)s + a² + b².
How much material do I need for a lampshade?
A lampshade has open ends, so use the lateral area π(R + r)s, then add a seam allowance.