Sphere

A sphere is a three-dimensional version of a circle, defined as the set of points equidistant from a central point. The distance from the center to any point on the sphere is called the radius (r). While a sphere is a hollow surface, a ball includes the space inside it. Both share the same radius, center, and diameter. The volume of a sphere is calculated as:

Volume = (4/3)πr³

Example: Claire wants to fill a spherical water balloon with a radius of 0.15 feet with vinegar. The required volume is:

Volume = (4/3) × π × 0.15³ = 0.141 ft³

Cone

A cone tapers smoothly from a circular base to a single point called the apex. The volume of a cone is given by:

Volume = (1/3)πr²h

where r is the base radius and h is the height.

Example: Bea compares the volumes of a waffle cone (radius 1.5 inches, height 5 inches) and a sugar cone. The waffle cone’s volume is:

Volume = (1/3) × π × 1.5² × 5 = 11.781 in³

Cube

A cube is a three-dimensional shape with six square faces, all meeting at right angles. Its volume is calculated as:

Volume = a³

where a is the length of an edge.

Example: Bob has a cubic suitcase with edges of 2 feet. Its volume is:

Volume = 2³ = 8 ft³

Cylinder

A cylinder consists of two parallel circular bases connected by a curved surface. Its volume is:

Volume = πr²h

where r is the radius and h is the height.

Example: Caelum uses cylindrical barrels (radius 3 feet, height 4 feet) to build a sandcastle. Each barrel’s volume is:

Volume = π × 3² × 4 = 113.097 ft³

Rectangular Tank

A rectangular tank is a box-like shape with six rectangular faces. Its volume is:

Volume = length × width × height

Example: Darby packs a rectangular bag (4 ft × 3 ft × 2 ft) with cake. The volume is:

Volume = 4 × 3 × 2 = 24 ft³

Capsule

A capsule consists of a cylinder with hemispherical ends. Its volume combines the formulas for a cylinder and a sphere:

Volume = πr²h + (4/3)πr³

Example: Joe fills a capsule (radius 1.5 feet, height 3 feet) with melted chocolate. The volume is:

Volume = π × 1.5² × 3 + (4/3) × π × 1.5³ = 35.343 ft³

Spherical Cap

A spherical cap is a portion of a sphere cut off by a plane. Its volume is:

Volume = (1/3)πh²(3R - h)

where R is the sphere’s radius and h is the cap’s height.

Example: Jack cuts a spherical cap (height 0.3 inches) from a golf ball (radius 1.68 inches). The volume is:

Volume = (1/3) × π × 0.3² × (3 × 1.68 - 0.3) = 0.447 in³

Conical Frustum

A conical frustum is the portion of a cone between two parallel planes. Its volume is:

Volume = (1/3)πh(r² + rR + R²)

where r and R are the radii of the two bases.

Example: Bea has a frustum (height 4 inches, radii 0.2 and 1.5 inches) of ice cream. The volume is:

Volume = (1/3) × π × 4 × (0.2² + 0.2 × 1.5 + 1.5²) = 10.849 in³

Ellipsoid

An ellipsoid is a stretched or compressed sphere. Its volume is:

Volume = (4/3)πabc

where a, b, and c are the lengths of the semi-axes.

Example: Xabat fills an ellipsoid bun (axes 1.5, 2, and 5 inches) with meat. The volume is:

Volume = (4/3) × π × 1.5 × 2 × 5 = 62.832 in³

Square Pyramid

A square pyramid has a square base and four triangular faces meeting at an apex. Its volume is:

Volume = (1/3)a²h

where a is the base edge length and h is the height.

Example: Wan builds a mud pyramid (base 5 feet, height 12 feet). The volume is:

Volume = (1/3) × 5² × 12 = 100 ft³

Tube

A tube is a hollow cylinder. Its volume is calculated by subtracting the inner cylinder’s volume from the outer one:

Volume = π(d₁² - d₂²)l / 4

where d₁ and d₂ are the outer and inner diameters, and l is the length.

Example: Beulah builds a pipe (outer diameter 3 feet, inner diameter 2.5 feet, length 10 feet). The volume is:

Volume = π × (3² - 2.5²) × 10 / 4 = 21.6 ft³