How to use it
- Choose Truncated cone or Square pyramid frustum.
- Enter the base (bottom) radius R or base side a, and the top radius r or top side b. Use the radius, not the diameter.
- Enter the perpendicular height h between the two faces.
- Read the volume, slant height and areas. All values use your length unit: cm in, cm³ and cm² out.
Formulas
Truncated cone (radii R and r, height h):
- Volume V = πh(R² + Rr + r²) / 3
- Slant height s = √(h² + (R − r)²)
- Lateral area = π(R + r)·s; base πR², top πr²
Square pyramid frustum (sides a and b):
- Volume V = h(A₁ + A₂ + √(A₁A₂)) / 3 with A₁ = a², A₂ = b²
- Slant height of a face s = √(h² + ((a − b)/2)²)
- Lateral area = 2(a + b)·s; base a², top b²
Worked example
Take a cone frustum with R = 3, r = 1, h = 4: V = π × 4 × (9 + 3 + 1) / 3 = 52π/3 = 54.4543. The slant height is √(16 + 4) = 4.4721, the lateral area π × 4 × 4.4721 = 56.1985, the base 9π = 28.2743 and the top π = 3.1416, so the total surface is 87.6144.
A square frustum with a = 6, b = 4, h = 3 has V = 3 × (36 + 16 + 24) / 3 = 76 and a lateral area of 2 × 10 × √10 = 63.2456.
Tips
- Buckets and cups: measure the diameters at the top and bottom and halve them. The volume in cm³ divided by 1000 is litres.
- Which radius is “base” does not matter for the volume. It only labels which end face is the base area and which is the top.
- Diameter by mistake gives four times the area and volume; if the result looks far too big, check that first.