Truncated Cone
Calculate truncated cone (conical frustum) volume, total surface area, lateral area, top and base areas, and slant height from top radius, bottom radius, and vertical height. Free online 3D geometry calculator with formula breakdown for engineering and design.
About This Calculator
A truncated cone, also known as a conical frustum, is a fundamental 3D geometric shape formed by slicing off the top of a right circular cone with a cutting plane parallel to its base. This shape appears throughout our daily lives in objects like drinking cups, buckets, flower planters, traffic cones, and architectural columns. Our comprehensive truncated cone calculator computes all major geometric properties from just three measurements: top radius, bottom radius, and vertical height.
The calculator applies the standard frustum formulas: volume V = (\u03c0h/3)(R\u00b2 + Rr + r\u00b2), lateral surface area A_l = \u03c0(r + R)l, top circular area A_t = \u03c0r\u00b2, base area A_b = \u03c0R\u00b2, total surface area A = A_l + A_t + A_b, and slant height l = \u221a(h\u00b2 + (R - r)\u00b2). All formulas assume a right circular frustum where the top and bottom faces are parallel circles centered on the same vertical axis.
When the top radius equals the bottom radius, the shape reduces to a cylinder. When the top radius approaches zero, it becomes a full cone. This relationship makes the truncated cone a versatile intermediate shape between cylinders and cones, often used in transitional geometries for ductwork, piping reducers, and architectural column designs.
This tool is invaluable for students learning solid geometry, engineers designing tapered components, contractors estimating concrete volumes for tapered foundations, and manufacturers determining material requirements for conical products. The calculator supports URL parameter sharing, making it easy to save and share specific calculation configurations with colleagues or classmates. Whether you need to calculate the volume of a custom bucket, determine material needed for a tapered planter, or solve a geometry homework problem, this calculator provides accurate results instantly.
Frequently Asked Questions
What is a truncated cone?
A truncated cone, or conical frustum, is a cone with its top cut off by a plane parallel to the base. It has two circular faces of different radii (top and bottom) connected by a sloping lateral surface. Common examples include buckets, cups, planters, and tapered drinking glasses.
What formulas does the truncated cone calculator use?
The calculator uses: Volume V = (\u03c0h/3)(R\u00b2 + Rr + r\u00b2), Lateral area A_l = \u03c0(r + R)l, Top area A_t = \u03c0r\u00b2, Base area A_b = \u03c0R\u00b2, Total surface area A = A_l + A_t + A_b, and Slant height l = \u221a(h\u00b2 + (R - r)\u00b2).
What is the difference between slant height and vertical height?
Vertical height (h) is the perpendicular distance between the top and bottom circular faces measured straight up. Slant height (l) is the distance measured along the sloping lateral surface from the edge of the top circle to the edge of the bottom circle. They are related by l\u00b2 = h\u00b2 + (R - r)\u00b2.
How many faces does a truncated cone have?
A truncated cone has three faces: two circular faces (the top and bottom bases) and one curved lateral surface. The two circular faces are parallel to each other but have different radii. The curved lateral surface connects the edges of both circular faces.
Where are truncated cone calculations used in real life?
Truncated cone calculations are used in manufacturing for designing buckets, cups, funnels, and tapered containers; in construction for concrete columns and hopper volumes; in architecture for decorative pillars; in the food industry for cup and container sizing; and in engineering for fluid flow calculations through tapered pipes.
Can I use different units for radius and height measurements?
All measurements must be in the same unit for accurate results. If you enter radii in inches, height must also be in inches. The calculator does not perform unit conversion. Convert all measurements to the same unit before entering them using our unit conversion tools if needed.
Is the truncated cone calculator free?
Yes, this calculator is completely free with no registration or limits. Use it for unlimited calculations and share results via URL for educational, professional, or personal use.