Volume of Solids

Volume measures the amount of three-dimensional space enclosed within a solid shape, expressed in cubic units such as cm³, m³, or in³. Among the most important solids in mathematics and science are the cone and the sphere — shapes that appear everywhere from ice-cream scoops to planetary bodies. Understanding how to calculate their volumes unlocks problems in engineering, architecture, physics, and everyday life. The volume of a cone is exactly one-third the volume of a cylinder that shares the same base radius and height. This elegant relationship was first proven rigorously by the ancient Greek mathematician Archimedes, who also famously showed that the volume of a sphere equals two-thirds the volume of its circumscribed cylinder. These results were so important to Archimedes that he requested a sphere-and-cylinder be engraved on his tombstone. In this article you will derive and apply the formulas V = (1/3)πr²h for cones and V = (4/3)πr³ for spheres. You will see worked examples with real numbers, understand where the formulas come from through Cavalieri's Principle, and connect these ideas to related concepts such as surface area and the volumes of cylinders and pyramids.

Key takeaways

Review: Volume of a Cylinder
Volume of a Cone — the One-Third Rule
Volume of a Sphere — the Four-Thirds Rule
The Hemisphere Connection
Choosing the Right Dimensions

Review: Volume of a Cylinder

Before tackling cones, recall that a right circular cylinder with base radius r and height h has volume V = πr²h. This is simply the area of the circular base (πr²) multiplied by the height. All cone and sphere formulas are built on top of this foundation.

Volume of a Cone — the One-Third Rule

A right circular cone with base radius r and perpendicular height h has volume V = (1/3)πr²h. The factor of 1/3 can be understood intuitively by noting that exactly three identical cones can be packed (without overlap) to fill one cylinder of the same r and h. A rigorous proof uses Cavalieri's Principle: every horizontal cross-section of the cone at height y from the apex is a circle whose radius scales linearly as r·(y/h), giving area π(ry/h)². Integrating from 0 to h yields ∫₀ʰ π(r²y²/h²) dy = πr²/h² · [y³/3]₀ʰ = (1/3)πr²h.

Volume of a Sphere — the Four-Thirds Rule

A sphere of radius r has volume V = (4/3)πr³. Archimedes proved this using the method of exhaustion; modern calculus confirms it by integrating circular cross-sections. At height y above the centre, a horizontal slice has radius √(r²−y²) and area π(r²−y²). Integrating from −r to r: V = ∫₋ᵣʳ π(r²−y²) dy = π[r²y − y³/3]₋ᵣʳ = π[(r³ − r³/3) − (−r³ + r³/3)] = π · (4r³/3) = (4/3)πr³.

The Hemisphere Connection

A hemisphere (half-sphere) simply has volume V = (1/2)·(4/3)πr³ = (2/3)πr³. Notice this equals exactly the volume of a cylinder with the same radius and height r minus the cone of radius r and height r: πr³ − (1/3)πr³ = (2/3)πr³. This is the precise relationship Archimedes discovered and treasured.

Choosing the Right Dimensions

Always identify r (radius) and h (perpendicular height, not slant height) before substituting into the cone formula. For a sphere, only r is needed. Common errors include using the slant length ℓ = √(r²+h²) instead of h, or forgetting the (1/3) factor. Slant height is used only for surface area calculations, not volume.

The Key Equation

V_{\text{cone}} = \dfrac{1}{3}\pi r^{2}h \qquad V_{\text{sphere}} = \dfrac{4}{3}\pi r^{3}

V — Volume (cm³ (or m³, in³, etc.)): The amount of three-dimensional space enclosed by the solid.
r — Radius (cm (or any length unit)): For a cone, the radius of the circular base. For a sphere, the radius from the centre to the surface.
h — Perpendicular Height (cm (or any length unit)): The vertical distance from the base of the cone to its apex, measured at a right angle to the base. Not used for spheres.
\pi — Pi (dimensionless): The mathematical constant approximately equal to 3.14159, relating a circle's circumference to its diameter.

Worked Example

Problem. A waffle cone has a base diameter of 8 cm and a perpendicular height of 15 cm. An ice-cream scoop sits on top in the shape of a perfect sphere of diameter 8 cm. Calculate (a) the volume of ice cream inside the cone, and (b) the volume of the spherical scoop. Give answers to 3 significant figures.

V_{\text{cone}} = 80\pi \approx 251\text{ cm}^{3}, \quad V_{\text{sphere}} = \dfrac{256\pi}{3} \approx 268\text{ cm}^{3}

Answer. Volume of cone ≈ 251 cm³; Volume of spherical scoop ≈ 268 cm³

Figure

Diagram

Origins · 1 minArchimedes of Syracuse and how this was discovered

Archimedes of Syracuse

-287–-212 · Greek (Sicilian)

Archimedes proved the volumes of the sphere and cone using his method of exhaustion — an early precursor to integral calculus. He showed that V_sphere = (2/3)V_cylinder and V_cone = (1/3)V_cylinder for solids with equal radii and height. His work 'On the Sphere and Cylinder' remains one of the greatest achievements of ancient mathematics. Archimedes was so proud of his sphere-cylinder result that he asked for a diagram of a sphere inside a cylinder to be engraved on his tomb. The Roman statesman Cicero later reported finding the tomb and recognising it by this very engraving, over 130 years after Archimedes' death.

Volume of a Cylinder — The cone formula V = (1/3)πr²h is exactly one-third of the cylinder formula V = πr²h. Understanding cylinders is essential before studying cones.
Surface Area of Cones and Spheres — Surface area and volume are complementary measurements of the same solids. The slant height used in cone surface area is distinct from the perpendicular height used in volume.
Volume of a Pyramid — A pyramid also has volume = (1/3) × base area × height, showing that the one-third rule applies to any pointed solid, not just cones.
Integration to Find Volumes of Revolution — The derivation of both formulas using slicing and integration is the foundation of volumes of revolution in calculus, where any solid of revolution can be calculated.