Volume (geometry)
Volume is the measure of the three-dimensional space enclosed by a closed surface, expressed in cubic units. It quantifies the capacity or amount of space a solid object occupies, and is essential in engineering, physics, chemistry, and everyday applications like packaging and fluid storage. Each solid shape has a specific volume formula derived from its geometry, such as V = lwh for a cuboid or V = (4/3)πr³ for a sphere.
Key Formula
V_cuboid = l × w × h; V_sphere = (4/3)πr³; V_cylinder = πr²h
LaTeX: V_{\text{cuboid}} = l \cdot w \cdot h, \quad V_{\text{sphere}} = \dfrac{4}{3}\pi r^3, \quad V_{\text{cylinder}} = \pi r^2 h
| Symbol | Meaning | Unit |
|---|---|---|
| l | Length of cuboid | m |
| w | Width of cuboid | m |
| h | Height | m |
| r | Radius of sphere or cylinder | m |
Worked Example
Problem
A cylindrical water tank has a radius of 3 m and a height of 5 m. How many litres of water can it hold? (1 m³ = 1000 L)
Solution
Step 1 — Volume formula: V = πr²h. Step 2 — Substitute: V = 3.14159 × 3² × 5 = 3.14159 × 9 × 5 = 3.14159 × 45 ≈ 141.37 m³. Step 3 — Convert: 141.37 m³ × 1000 L/m³ = 141,372 L.
Answer
Volume ≈ 141.37 m³ ≈ 141,372 litres
Volume Formulas for Common 3D Solids
| Solid | Volume Formula | Key Variables | Example (units in cm) |
|---|---|---|---|
| Cube | V = s³ | s = side | s=4 → V = 64 cm³ |
| Cuboid | V = l×w×h | l, w, h = dimensions | l=5,w=3,h=2 → V=30 cm³ |
| Cylinder | V = πr²h | r = radius, h = height | r=3,h=7 → V≈197.92 cm³ |
| Sphere | V = (4/3)πr³ | r = radius | r=5 → V≈523.6 cm³ |
| Cone | V = (1/3)πr²h | r = base radius, h = height | r=4,h=9 → V≈150.8 cm³ |
| Pyramid | V = (1/3)Bh | B = base area, h = height | B=25,h=6 → V=50 cm³ |
Interactive Tools
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Related Terms
Surface Area
Surface area is the total area of all the outer faces or surfaces of a three-dimensional solid, expressed in square units. It measures how much material is needed to cover an object completely and is critical in applications such as packaging design, heat transfer calculations, and chemical reaction rates (which depend on exposed surface area). For a closed solid, the surface area is found by summing the areas of every face or, for curved surfaces, by integration.
Area (geometry)
Area is the measure of the two-dimensional region enclosed within a closed geometric figure, expressed in square units. It quantifies how much flat surface a shape covers and is fundamental in fields ranging from architecture and land surveying to physics and engineering. Different shapes have distinct area formulas derived from their geometric properties, such as A = πr² for a circle or A = ½bh for a triangle.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is the study of geometric figures using a coordinate system in which points are represented by numerical coordinates. It provides a powerful bridge between algebra and geometry, allowing geometric shapes to be described by equations and algebraic operations to yield geometric insights. Key concepts include plotting points, finding distances, midpoints, slopes, and the equations of lines, circles, and conic sections on the Cartesian plane.
From Latin "volumen" meaning "a roll" or "a coil" (from "volvere", to roll), later generalised to mean "bulk" or "size". Its mathematical sense of three-dimensional measure was formalised by Archimedes of Syracuse (c. 287–212 BCE), who derived volume formulas for spheres and cylinders.