Cone and Pyramid Calculator
Compute volume, total surface area, or slant height for cones, square pyramids, and triangular pyramids. Useful for geometry homework, 3D printing projects, and architecture.
About this calculator
Cones and pyramids share a common volume formula: V = (1/3) × B × h, where B is the base area and h is the perpendicular height. For a cone, B = π × r², giving V = (1/3) × π × r² × h. For a square pyramid, B = s², so V = (1/3) × s² × h. The slant height l is found using the Pythagorean theorem: l = √(r² + h²) for a cone or l = √(s² + h²) for a pyramid. Total surface area of a cone is π × r × (r + l), while a square pyramid's total surface area is s² + 2 × s × l. These formulas apply to right cones and right pyramids, where the apex sits directly above the center of the base.
How to use
Suppose you have a square pyramid with base side s = 6 units and height h = 4 units. First, compute slant height: l = √(6² + 4²) = √(36 + 16) = √52 ≈ 7.21 units. Volume: V = (1/3) × 6² × 4 = (1/3) × 36 × 4 = 48 cubic units. Total surface area: SA = 6² + 2 × 6 × 7.21 = 36 + 86.57 ≈ 122.57 square units. Select 'Square Pyramid', enter base dimension = 6, height = 4, and pick your desired output.
Frequently asked questions
What is the difference between slant height and vertical height of a pyramid?
Vertical height (h) is the perpendicular distance from the apex straight down to the center of the base. Slant height (l) is the distance from the apex down the face to the midpoint of a base edge. They are related by the Pythagorean theorem: l = √((s/2)² + h²) for a square pyramid or l = √(r² + h²) for a cone. Slant height is needed when calculating lateral surface area, while vertical height is used in the volume formula.
How do I calculate the volume of a cone with radius 5 and height 9?
Use the cone volume formula V = (1/3) × π × r² × h. Substituting r = 5 and h = 9 gives V = (1/3) × π × 25 × 9 = (1/3) × 225π ≈ 235.62 cubic units. The factor of one-third distinguishes a cone from a cylinder of the same base and height, which would have three times the volume. This relationship holds for all pyramids and cones regardless of base shape.
Why does a pyramid have one-third the volume of a prism with the same base and height?
This result follows from Cavalieri's principle and can be proved by dissecting a prism into exactly three congruent pyramids. Intuitively, as you stack infinitely thin cross-sections from base to apex, each horizontal slice shrinks quadratically, integrating to exactly one-third the total prism volume. The formula V = (1/3) × B × h applies to any pyramid or cone, no matter the shape of the base. It is one of the fundamental results of solid geometry, first rigorously proved by Eudoxus of Cnidus around 350 BCE.