Frustum Calculator
Calculate volume and surface area of a frustum (truncated cone or pyramid). Get slant height, lateral area, and full cone relationship calculations.
Frustum Calculator
Calculator
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Formula: V = (pi h / 3)(R^2 + Rr + r^2) | Lateral Area = pi(R + r) x slant height
Worked example — Volume: 17,247.8 cm^3 (17.25 L) | Total Surface Area: 3,716.7 cm^2
Formula
V = (pi h / 3)(R^2 + Rr + r^2) | Lateral Area = pi(R + r) x slant height
Where V = volume, h = perpendicular height of the frustum, R = radius of the larger base, r = radius of the smaller base, and slant height = sqrt(h^2 + (R-r)^2). For a square pyramid frustum, replace circular areas and perimeters with square equivalents.
Worked Examples
Example 1: Bucket Volume Calculation
Problem:A bucket has a bottom radius of 12 cm, top radius of 15 cm, and height of 30 cm. Calculate the volume and total surface area.
Solution:Volume = (pi x 30 / 3) x (12^2 + 12 x 15 + 15^2) = (10pi) x (144 + 180 + 225) = 10pi x 549 = 5490pi = 17,247.8 cm^3 = 17.25 liters Slant height = sqrt(30^2 + (12-15)^2) = sqrt(900 + 9) = 30.15 cm Lateral area = pi x (12 + 15) x 30.15 = 814.04pi = 2,557.4 cm^2 Top area = pi x 15^2 = 225pi = 706.9 cm^2 Bottom area = pi x 12^2 = 144pi = 452.4 cm^2 Total = 3,716.7 cm^2
Result:Volume: 17,247.8 cm^3 (17.25 L) | Total Surface Area: 3,716.7 cm^2
Example 2: Concrete Footing Volume
Problem:A frustum-shaped concrete footing has a 4 ft bottom radius, 2 ft top radius, and 3 ft height. Calculate cubic yards of concrete needed.
Solution:Volume = (pi x 3 / 3) x (4^2 + 4 x 2 + 2^2) = pi x (16 + 8 + 4) = 28pi = 87.96 ft^3 Convert to cubic yards: 87.96 / 27 = 3.26 cubic yards Slant height = sqrt(3^2 + (4-2)^2) = sqrt(9 + 4) = 3.61 ft Lateral area = pi x (4 + 2) x 3.61 = 21.66pi = 68.04 ft^2
Result:Volume: 87.96 ft^3 (3.26 cubic yards) | Lateral Area: 68.04 ft^2
Frequently Asked Questions
What is a frustum and how is it formed?
A frustum is a three-dimensional geometric solid created by cutting a cone or pyramid with a plane parallel to its base. This cut removes the top portion (the apex region) and leaves a shape with two parallel faces of different sizes connected by a sloping lateral surface. The word frustum comes from the Latin word meaning 'morsel cut off.' Frustums appear frequently in everyday life: drinking cups, buckets, lampshades, and architectural columns are all examples of conical frustums. The frustum retains the symmetry properties of its parent shape, meaning a frustum of a right circular cone has a circular cross-section at every height.
How is the volume of a frustum calculated?
The volume of a frustum of a cone is calculated using the formula V = (pi times h divided by 3) times (R squared plus R times r plus r squared), where h is the height, R is the radius of the larger base, and r is the radius of the smaller base. This formula can be derived by subtracting the volume of the removed small cone from the volume of the original full cone. For a frustum of a square pyramid, the formula is V = (h/3)(A1 + A2 + sqrt(A1 times A2)), where A1 and A2 are the areas of the two parallel faces. Both formulas reduce to the full cone or pyramid volume formula when the top radius or area equals zero.
What is the slant height of a frustum and how is it found?
The slant height of a frustum is the shortest distance measured along the lateral (sloping) surface from the edge of one base to the corresponding edge of the other base. For a conical frustum, the slant height is calculated using the Pythagorean theorem: slant height equals the square root of (h squared plus (R minus r) squared), where h is the perpendicular height and R and r are the bottom and top radii. This relationship exists because if you unfold a vertical cross-section of the frustum, the height, the radius difference, and the slant height form a right triangle. The slant height is essential for calculating the lateral surface area of the frustum.
How is the lateral surface area of a frustum computed?
The lateral surface area of a conical frustum is the area of the curved surface that connects the two circular bases, excluding the top and bottom faces. The formula is Lateral Area = pi times (R plus r) times the slant height, where R is the larger radius, r is the smaller radius, and the slant height equals sqrt(h squared plus (R minus r) squared). This formula can be understood geometrically: if you unroll the lateral surface, it forms a flat shape called an annular sector, and pi times (R plus r) represents the average circumference of the two bases. The total surface area is found by adding the areas of both circular bases to the lateral area.
What are common real-world applications of frustum calculations?
Frustum calculations are used extensively in engineering, architecture, and manufacturing. In civil engineering, dam cross-sections, retaining walls, and foundation footings are often frustum-shaped, requiring volume calculations for material estimation. In manufacturing, tapered containers like buckets, cups, and funnels require frustum geometry for determining material usage and capacity. Architectural features like tapered columns, cooling towers, and spires use frustum mathematics for structural design. In agriculture, grain storage bins and water tanks are frequently frustum-shaped. Even in optics and lighting, the beam spread of a flashlight or projector creates a frustum shape, and calculating illumination requires understanding frustum geometry.
How does a frustum relate to its parent cone or pyramid?
A frustum can always be related back to its parent cone or pyramid through geometric similarity. If the frustum has bottom radius R, top radius r, and height h, the full height of the original cone would be H = hR/(R-r), derived from similar triangles. The removed apex portion has height H-h = hr/(R-r). The volume of the frustum equals the volume of the full cone minus the volume of the removed cap. This relationship is useful for deriving frustum formulas and for understanding proportionality. For example, a frustum created by cutting a cone at half its height retains 7/8 of the original volume, not 1/2 as many people intuitively guess, because volume scales with the cube of linear dimensions.
What is the difference between a right frustum and an oblique frustum?
A right frustum has its axis perpendicular to both bases, meaning the line connecting the centers of the two parallel faces is at a 90-degree angle to those faces. This is the standard frustum shape most people visualize. An oblique frustum has its axis tilted at an angle, so while the two bases are still parallel, the connecting axis is not perpendicular to them. The standard frustum volume formula works for both right and oblique frustums by Cavalieri principle, since corresponding cross-sectional areas at equal heights are identical regardless of tilt. However, the surface area calculations differ for oblique frustums because the slant height varies around the circumference, making lateral area computations more complex.
How do you calculate the frustum dimensions from a full cone and cutting height?
If you have a full cone with base radius R and height H, and you cut it at height h from the base (removing the top), you can find all frustum dimensions using similar triangles. The top radius of the frustum is r = R times (H - h) / H, because the radius scales linearly with distance from the apex. The frustum height is simply h. From these three values, you can compute everything: volume = (pi times h / 3)(R squared + Rr + r squared), slant height = sqrt(h squared + (R-r) squared), and all surface areas. For example, cutting a cone with R = 10 and H = 20 at h = 15 gives r = 10 times 5/20 = 2.5, so the frustum has radii 10 and 2.5 with height 15.
Can the frustum formula be extended to higher-dimensional shapes?
Yes, the prismatoid formula generalizes the frustum volume concept to any solid bounded by two parallel planes. The prismatoid formula states V = (h/6)(A1 + 4Am + A2), where A1 and A2 are the areas of the two parallel end faces and Am is the area of the cross-section at the midpoint. This is equivalent to applying Simpson rule for numerical integration. The standard frustum formula is a special case of the prismatoid formula. In higher dimensions, analogous formulas exist for hyper-frustums. The prismatoid formula also applies to shapes that are not frustums, such as prisms, pyramids, wedges, and other solids with parallel end faces, making it one of the most versatile volume formulas in geometry.
How accurate are frustum approximations for irregular tapered shapes?
Using frustum formulas to approximate the volume of irregular tapered objects provides surprisingly good accuracy when the taper is roughly linear. The frustum formula is exact for cones and pyramids, and serves as an excellent approximation for shapes like tree trunks, tapered pipes, and beverage containers that deviate slightly from perfect conic form. For better accuracy with irregular shapes, the object can be divided into multiple thin frustum sections (a technique called frustum stacking or disk integration), with each section using locally measured top and bottom dimensions. This approach is mathematically equivalent to numerical integration and is used in forestry for timber volume estimation, in geology for calculating volumes of sedimentary deposits, and in manufacturing for quality control of machined parts.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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