🔄 Revolved Trapezoid Calculator
Volume of a trapezoid rotated around an external axis (solid of revolution)
Trapezoid Dimensions
Axis Position
Trapezoid rotated 360° around axis
📊 Revolved Trapezoid Results
Top Width (a)
4 cm
Bottom Width (b)
8 cm
Height (h)
6 cm
Inner Radius r₁
5 cm
Outer Radius r₂
11 cm
Trapezoid Area
36 cm²
Centroid Distance
7.56 cm
Volume
1,710 cm³
📝 Step-by-Step Solution (Pappus Theorem)
Given: a = 4 cm, b = 8 cm, h = 6 cm, r₁ = 5 cm, r₂ = 11 cm
Trapezoid area: A = ((a + b) / 2) × h = ((4 + 8) / 2) × 6 = 36 cm²
Centroid distance from axis: ȳ = r₁ + ((h(2b + a)) / (3(a + b))) =
7.56 cm
Pappus theorem: V = 2π × ȳ × A = 2π × 7.56 × 36 = 1,710 cm³
Alternative verify: V = (π/3) × h × (r₂²(2b+a) - r₁²(2a+b)) / (but
Pappus is elegant!)
📐 Revolved Trapezoid Formulas
Pappus Theorem: V = 2π × ȳ × A
Trapezoid Area: A = ((a + b) / 2) × h
Centroid from axis: ȳ = r₁ + (h(2b + a)) / (3(a + b))
Surface Area: SA = 2π × ȳ × Perimeter
Understanding Revolved Trapezoids
🔄 Solid of Revolution
Rotating a 2D shape around an axis creates a 3D solid. A revolved trapezoid creates a donut-like or frustum-ring shape.
📐 Pappus Theorem
V = 2π × centroid distance × Area. The elegant theorem: volume equals path traveled by centroid times cross-section area.
⭕ External Axis
Axis outside the shape. Creates a hollow ring-like solid. If axis touches shape, it's a cone or frustum.
📍 Centroid
Center of mass of the trapezoid. Located at ȳ from the axis. Key to Pappus calculation.
Frequently Asked Questions
What is a revolved trapezoid?
A solid of revolution. Take a trapezoid cross-section and rotate
it 360° around an external axis. Creates a ring or donut-like shape with trapezoidal cross-section.
What is Pappus theorem?
V = 2π × ȳ × A. Volume equals 2π times the centroid distance from
axis times the area. Named after Pappus of Alexandria (4th century).
How do I find the centroid of a trapezoid?
ȳ = r₁ + (h(2b + a)) / (3(a + b)). Distance from axis to
trapezoid's center of mass. Depends on the two parallel sides and height.
What is an external axis?
Axis of rotation outside the shape. The trapezoid doesn't touch
the axis. Creates a hollow torus-like solid rather than a simple cone.
What shape does revolved trapezoid make?
Toroidal frustum or annular wedge. Like a slice of a donut with
trapezoidal cross-section. Ring-shaped with varying thickness.
How is this different from a cone frustum?
Axis position. Frustum = axis through the shape. Revolved
trapezoid = axis outside, creating a hollow ring structure.
What are r₁ and r₂?
Inner and outer radii. r₁ = distance from axis to nearest edge. r₂
= distance to farthest edge. r₂ - r₁ = height h for perpendicular orientation.
Can I use this for partial rotation?
Multiply by angle fraction. For 180° rotation, volume = half. For
any angle θ: V = (θ/360°) × full volume.
What if the axis touches the trapezoid?
Set r₁ = 0. This creates a solid frustum or cone. Pappus still
works, but there's no hollow center.
How do I find surface area?
Pappus for surfaces: SA = 2π × ȳ × P. Where P is the perimeter of
the trapezoid, not the area.
What is the slant height?
s = √(h² + ((b-a)/2)²). The non-parallel sides of the trapezoid.
Needed for perimeter calculations.
What are real-world applications?
Pipe fittings, turbine blades, flywheel segments, ring gaskets.
Any ring-shaped object with tapered cross-section.
Why use Pappus instead of integration?
Simplicity. Pappus reduces complex integration to simple
multiplication. Just need area and centroid location.
How accurate is this calculator?
Mathematically exact. Uses Pappus theorem with precise centroid
formula. Same result as disk/washer integration.
What if a > b (inverted trapezoid)?
Still works. The formulas handle both orientations. Centroid
position adjusts automatically.
Can I calculate mass from volume?
Mass = Volume × Density. Find volume first, then multiply by
material density (g/cm³ or kg/m³).
What is the moment of inertia?
More complex calculation. Requires integration or specialized
formulas. Used for rotational dynamics of the solid.