Volume Calculator

A rectangular prism, also known as a cuboid or rectangular box, is a three-dimensional geometric shape with six rectangular faces, twelve edges, and eight vertices. Each face is a rectangle, and opposite faces are parallel and congruent. The three dimensions (length, width, height) are mutually perpendicular, creating right angles at every corner.

The rectangular prism volume formula (V = l × w × h) represents the most fundamental volume calculation in geometry. It directly applies the concept that volume equals the product of three perpendicular dimensions. This shape is the most common in manufactured objects because it's efficient for storage, transportation, and construction. Examples include boxes, rooms, books, refrigerators, and most building structures.

A sphere is a perfectly round three-dimensional geometric object where every point on its surface is equidistant from its center. This distance is called the radius. The sphere is the three-dimensional analog of a circle and has the smallest surface area among all surfaces that enclose a given volume, making it an efficient shape in nature and engineering.

The sphere volume formula involves π (pi ≈ 3.14159) because it relates to circular geometry. The factor 4/3 comes from calculus integration of circular cross-sections. To calculate, cube the radius, multiply by π, then multiply by 4/3. Spheres appear in balls, planets, bubbles, and many natural formations like dewdrops and cellular structures.

A cylinder is a three-dimensional geometric shape consisting of two parallel circular bases of equal size connected by a curved rectangular surface. It can be thought of as a circle extended through space in a perpendicular direction. Cylinders are classified as right cylinders (where the sides are perpendicular to the base) or oblique cylinders (where the sides are at an angle).

The cylinder volume formula (V = πr²h) works by calculating the area of the circular base (πr²) and multiplying it by the height. This represents stacking circular cross-sections from bottom to top. The π factor accounts for the circular base geometry. Cylinders are ubiquitous in engineering applications including pipes, tanks, cans, barrels, and engine cylinders.

A cone is a three-dimensional geometric shape that tapers smoothly from a flat circular base to an apex (vertex or tip). The apex lies directly above the center of the base in a right circular cone. The curved surface connecting the base to the apex is called the lateral surface, and the straight line from the apex to the center of the base represents the height.

The cone volume formula (V = ⅓πr²h) is exactly one-third of a cylinder with the same base and height. This 1/3 factor comes from the fact that a cone occupies one-third the space of its corresponding cylinder. The mathematical relationship is derived through calculus integration of circular cross-sections that decrease linearly from base to apex. Cones appear in ice cream cones, traffic cones, volcanic formations, and coniferous tree shapes.

A triangular prism is a three-dimensional geometric shape with two parallel triangular bases connected by three rectangular faces. The triangular bases are congruent and parallel to each other, while the rectangular faces form the lateral surfaces. This shape can be visualized as a triangle extruded through space in a perpendicular direction.

The triangular prism volume formula (V = ½ × b × h × l) works by first calculating the area of the triangular base (½ × base × height) and then multiplying by the length of the prism. This shape is commonly found in architectural elements like roof trusses, tent structures, chocolate bars (Toblerone), and various engineering applications where triangular cross-sections provide structural efficiency.

An ellipsoid is a three-dimensional geometric shape that resembles a stretched or compressed sphere. It has three perpendicular axes of different lengths, making it an oval-shaped solid. When all three axes are equal, the ellipsoid becomes a sphere. Ellipsoids are classified as prolate (elongated like a football) or oblate (flattened like Earth).

The ellipsoid volume formula (V = 4/3πabc) is an extension of the sphere formula, where instead of using radius³, we multiply the three semi-axes lengths. This shape appears in nature as eggs, planets (Earth is an oblate ellipsoid), rugby balls, and many biological structures like cells and organs.

A capsule, also known as a spherocylinder, is a three-dimensional shape consisting of a cylinder with hemispherical ends. It combines the cylindrical middle section with two half-spheres at each end, creating a smooth, rounded shape with no sharp edges. The total length includes both the cylindrical portion and the two hemispheres.

The capsule volume formula combines a cylinder volume (πr²h) with a complete sphere volume (4/3πr³). The two hemispheres together form one complete sphere. This shape is commonly found in pharmaceutical capsules, storage tanks, pressure vessels, and architectural elements where smooth, rounded ends are desired for safety or aesthetic reasons.

A conical frustum, also called a truncated cone, is a three-dimensional shape created by cutting a cone with a plane parallel to its base, removing the top portion. This results in a shape with two circular bases of different sizes connected by a curved lateral surface. The bases are parallel but not equal in size.

The conical frustum volume formula accounts for the gradual change in radius from one base to the other. It's essentially the volume of the complete cone minus the volume of the removed top portion. This shape is commonly found in lampshades, buckets, flower pots, traffic cones, and many industrial containers where a tapered design is beneficial for stacking or pouring.

A hexagonal prism is a three-dimensional geometric shape with two parallel hexagonal bases connected by six rectangular faces. The hexagonal bases are regular hexagons (six equal sides and angles), making this a uniform prism. Each hexagonal base has six equal sides and six equal angles of 120 degrees.

The hexagonal prism volume formula first calculates the area of the hexagonal base using the formula (3√3/2)a², then multiplies by height. The factor 3√3/2 ≈ 2.598 comes from the geometric properties of a regular hexagon. This shape appears in nature as honeycomb cells, crystal structures, pencils, nuts and bolts, and various architectural elements.

A pentagonal prism is a three-dimensional geometric shape with two parallel pentagonal bases connected by five rectangular faces. The pentagonal bases are regular pentagons with five equal sides and five equal angles of 108 degrees each. This shape combines the unique properties of the pentagon with the structural stability of a prism.

The pentagonal prism volume formula involves calculating the area of the regular pentagonal base, which uses the complex expression (5/4) × a² × √(25 + 10√5) ≈ 1.720a², then multiplying by height. The pentagon appears in nature, architecture, and design, making pentagonal prisms useful in specialized applications like certain building designs and geometric art.