How is the volume of a sphere calculated?+
V = (4/3)πr³. For a sphere with radius 1 m: V = (4/3) × 3.14159 × 1³ = 4.189 m³. For radius 2 m: V = (4/3)π × 8 = 33.51 m³ (8× larger for 2× radius). Surface area = 4πr². Common applications: pressure vessels, tanks, ball bearings, and buoyancy calculations.
What is the formula for a cylinder?+
V = πr²h where r is base radius and h is height. Surface area = 2πr(r + h) = 2 circles + lateral surface. Example: r=0.5m, h=2m: V = π × 0.25 × 2 = 1.571 m³, SA = 2π(0.5)(0.5+2) = 7.854 m². Cylinder volume is the basis for pipe flow calculations, tank sizing, and column concrete requirements.
How is cone volume calculated and why is it 1/3?+
V = (1/3)πr²h. The 1/3 factor comes from the fact that if you fill a cone with sand and pour it into a same-base-and-height cylinder, it takes exactly 3 cones to fill. Archimedes demonstrated this geometrically. Slant height: l = √(r² + h²). Lateral surface area: πrl. Total surface: πr(r+l). Used for hopper sizing and funnel design.
What is the volume of a rectangular prism (box)?+
V = l × w × h. Surface area = 2(lw + lh + wh). Example: a room 5m × 4m × 3m: V = 60 m³ (for HVAC sizing). A shipping box 60cm × 40cm × 30cm: V = 72,000 cm³ = 0.072 m³ = 0.072 CBM. This is the most common volume formula in architecture, construction, and logistics.
How is a pyramid volume calculated?+
V = (1/3) × base area × height. For a square pyramid with side=4m and h=6m: base area = 16m², V = (1/3) × 16 × 6 = 32 m³. This formula applies to any pyramid regardless of base shape (square, rectangular, triangular). Like the cone, a pyramid is 1/3 of the corresponding prism with the same base and height.
What is the displacement method for irregular volumes?+
Submerge the object in a liquid-filled graduated container. Reading the rise in liquid level gives the displaced volume, which equals the object's volume. Archimedes used this to detect the goldsmith's fraud. Limitations: only works for non-porous, non-soluble, fully-submerged objects. Modern equivalent: 3D scanning with voxel counting for any shape.
What are CBM units in shipping?+
CBM = Cubic Meters, the standard freight volume unit in international shipping. 1 CBM = 1m × 1m × 1m = 1,000 liters. Freight pricing = max(actual weight, volumetric weight), where volumetric weight = CBM × 333 kg (sea) or CBM × 167 kg (air). A 40-foot shipping container holds about 67 CBM. Small parcels charging by dimensional weight use a similar volumetric formula.
How do surface area and volume scale with size?+
Volume scales as the cube of a linear dimension (V ∝ L³); surface area as the square (SA ∝ L²). Doubling all dimensions: volume ×8, surface area ×4. Consequence: large animals lose proportionally less heat (less SA/V ratio); cells divide when they grow too large (falling SA/V ratio reduces nutrient diffusion); large spherical tanks are more material-efficient per unit capacity.
What is a frustum and how is its volume calculated?+
A frustum is a cone or pyramid with the apex cut off parallel to the base, leaving two parallel bases. Volume = (h/3)(A₁ + A₂ + √(A₁ × A₂)) where A₁ and A₂ are the two base areas and h is height. Example: bucket with bottom radius 0.2m, top radius 0.3m, height 0.4m: A₁=πr₁²=0.1257, A₂=0.2827, V=(0.4/3)(0.1257+0.2827+0.1885)=0.0795 m³.
How is the volume of an ellipsoid calculated?+
V = (4/3)πabc where a, b, c are the three semi-axes. A sphere is a special case with a=b=c=r. The Earth is an oblate spheroid: a=b=6,378 km (equatorial), c=6,357 km (polar). Earth's volume ≈ 1.083×10¹² km³. Ellipsoids model: pressure vessels, eggs, Earth, and drug capsules. Surface area has no simple closed form (requires numerical integration).
What is the difference between volume and capacity?+
Volume = the 3D space any solid or void occupies, measured in m³ or ft³. Capacity = how much substance a container holds, typically in liters, gallons, or cubic feet. They measure the same physical quantity with different implied contexts. A water tank has a volume (the space the tank wall material occupies) and a capacity (the water volume it holds). In practice, the terms are often used interchangeably.
How is volume used in concrete calculation?+
Concrete is ordered by the cubic meter (metric) or cubic yard (imperial). A 4-inch thick slab: thickness=0.1016m. A 6m×4m slab: V = 6×4×0.1016 = 2.44 m³. Add 10% waste: order ≈ 2.7 m³. 1 cubic yard = 0.765 m³. A standard residential concrete truck carries ~8 cubic yards. For footings: V = length × width × depth. Always convert to consistent units before calculating.
How is volume measurement used in medicine?+
Medical applications: CT scan tumor volume quantification, lung volume spirometry (FEV₁, FVC in liters), cardiac stroke volume and ejection fraction (cardiac output = stroke volume × heart rate in L/min), medication dosing by body weight/volume distribution, infusion pump flow rates (mL/hour), and surgical implant sizing. Drug pharmacokinetics models use volume of distribution (L/kg) to relate blood concentration to actual body drug amount.