Frustum Cone Area - Slant and Total Area
Use this frustum cone area calculator to enter R, r, and H, then read the slant height, total surface area, and lateral area in cm, m, in, or ft.
Frustum Cone Area
Results
What Is a Frustum Cone Area?
A frustum cone area is the total surface area of a frustum of a cone, the solid that remains after you slice the top off a right circular cone with a plane parallel to the base. Use this frustum cone area calculator when you know the larger base radius R, the smaller top radius r, and the perpendicular height H, and need the total area, the lateral wall area, both circle areas, and the slant height.
- • Painting, coating, or finishing a planter or bucket: Compute the outside wall area of a tapered planter, waste bin, lampshade, or bucket to quote paint, stain, or epoxy coverage in square feet or square meters.
- • Sheet metal and material estimates: Estimate the metal or fabric needed to fabricate a conical transition piece, hopper, chute, or duct where the supplier quotes by the square unit of stock material.
- • Classwork and exam problems: Verify a hand calculation for the total or lateral area of a truncated cone, including limit cases r = 0 (full cone) or r = R (cylinder).
- • Capacity labels and product specs: Quote the lateral wall area and total area of a tapered container when the spec sheet needs an interior capacity and a finished surface measurement.
A frustum of a cone is a right circular cone with the tip cut off by a plane parallel to the base, leaving two parallel circular faces connected by a slanted wall. Flower pots, drinking glasses with a tapered base, lampshades, transition ducts, and the inside of a rocket nozzle are all close approximations of a right circular conical frustum.
If you also need the volume of the same frustum, the cone volume calculator uses the same R, r, and H inputs to return the capacity in your unit.
How the Frustum Cone Area Calculator Works
The calculator uses the closed-form formula A = pi * (R^2 + r^2 + S * (R + r)), where R is the larger base radius, r is the smaller top radius, and S is the slant height. It derives the slant height as S = sqrt(H^2 + (R - r)^2) and reports the total, lateral, and base areas in the unit you entered.
- R: Larger base radius. The radius of the bottom circle.
- r: Top radius. The radius of the upper circle. Use 0 for a full cone; must not exceed R.
- H: Perpendicular height between the two base planes.
- S: Slant height. Hypotenuse of a right triangle with vertical leg H and horizontal leg (R - r).
The slant height is the hypotenuse of a right triangle with vertical leg H and horizontal leg R - r. The lateral area comes from unrolling the slanted wall into a ring sector, giving the same pi * S * (R + r) closed form.
R=8 cm, r=2 cm, slant height 12 cm worked example
Larger base area = pi * 64 = 201.0619 cm^2. Top circle area = pi * 4 = 12.5664 cm^2. Lateral area = pi * 12 * 10 = 376.9911 cm^2. Total area = pi * 188 = 590.6195 cm^2.
Total = 590.6195 cm^2, lateral = 376.9911 cm^2, larger base = 201.0619 cm^2, top = 12.5664 cm^2, slant height = 12 cm.
Same total is reached from any R, r, H that produces S = 12, such as H = sqrt(108) ≈ 10.39 cm.
According to Wolfram MathWorld, the slant height is s = sqrt((R1 - R2)^2 + h^2) and the lateral surface area is pi * (R1 + R2) * s.
According to Wikipedia Frustum, the total surface area is pi * ((r1 + r2) * s + r1^2 + r2^2).
For users working with multiple solid shapes, the surface area calculator in the same math category covers spheres, prisms, pyramids, and frustums with consistent unit handling.
Key Concepts Behind the Frustum Cone Area
Four ideas show up every time you work with a truncated cone, and understanding each one makes the formula and the result panel easier to read.
Slant height S
S = sqrt(H^2 + (R - r)^2) is the distance from any point on the larger base edge to the corresponding point on the smaller top edge, measured along the slanted wall. It is the same triangle relationship that gives the slant of a complete cone.
Lateral vs total area
The lateral area is just the slanted wall, pi * S * (R + r). The total area adds the two base circles, pi * R^2 and pi * r^2, on top. If the open top is a hole rather than a sealed cap, the page can be read as lateral area only.
Closed-form total area formula
A = pi * (R^2 + r^2 + S * (R + r)) collapses three steps into one. It works for every right circular conical frustum, including the two limit cases r = 0 (full cone) and r = R (cylinder).
Square units and unit conversion
When the input is in centimeters, the area is in square centimeters. When the input is in feet, the area is in square feet. The page keeps a single unit selector so you do not have to convert the inputs or the outputs by hand.
When the design collapses into a cylinder because r = R, the cylinder volume calculator returns the surface area and volume of a right circular cylinder in the same unit.
How to Use the Frustum Cone Area Calculator
Run the calculator in five short steps, then read the result panel for the value you actually need.
- 1 Measure the larger base radius R: Measure or look up the radius of the larger base. If you only have a diameter, divide it by 2. The same unit will be used for r and H.
- 2 Measure the top radius r: Measure the radius of the smaller top circle. If the top is closed, measure to the inside of the rim. If the shape is a full cone, enter 0.
- 3 Measure the perpendicular height H: Measure the height from the larger base plane to the smaller top plane. Use a perpendicular measurement, not the slant length, so the slant height is derived correctly.
- 4 Pick the unit you measured in: Choose centimeters, meters, inches, or feet. The result panel switches to the matching square unit so you do not have to convert later.
- 5 Read the total area and supporting values: Use the total area for paint, fabric, sheet metal, or material quotes. Use the lateral area for the slanted wall only, the two base areas for caps or platforms, and the slant height for layout work.
A workshop is coating the outside of a tapered steel planter with R = 25 cm, r = 18 cm, and H = 30 cm. The operator enters those values in cm and reads slant height 31.40 cm, lateral area 4245.6 cm^2, larger base 1963.5 cm^2, top area 1017.9 cm^2, and total area 7226.9 cm^2 to size the paint order.
To compare the total area in cm^2 to a quote that comes in ft^2 or m^2, the area converter converts the square units without you squaring the conversion factor by hand.
Benefits of Using This Frustum Cone Area Calculator
The page is built for the way people use a frustum measurement in real work, with each benefit tied to a concrete decision or workflow.
- • Three inputs cover every truncated cone: Enter R, r, and H, and the page returns the slant height, total area, lateral area, larger base, and top circle in the same unit, so there is no second measurement to take to get a complete answer.
- • Works in four common length units: Centimeters, meters, inches, and feet cover most craft, school, fabrication, and engineering tasks without manual unit conversion, and the area output is labeled with the matching square unit.
- • Handles the two limit cases automatically: Entering r = 0 returns the surface area of a complete cone, and entering r = R returns the surface area of a cylinder. The same page replaces a cone area tool and a cylinder area tool for the limit shapes.
- • Shows the breakdown, not only the total: Lateral area, larger base, top circle, and slant height are on the same screen, so you can quote a wall-only paint job, a sealed cap, or a partial coating without re-running the calculation.
- • Cites the formula source: The page links to a Wolfram MathWorld reference, a Wikipedia Frustum article, and the Omni Calculator worked example, so the result is traceable for classwork or technical documentation.
When the project moves from a round truncated cone to a six-sided truncated pyramid, the hexagonal pyramid surface area calculator returns the area of each face and the total for the polyhedral version.
Factors That Affect the Frustum Cone Area Result
The closed-form formula does not have hidden variables, but the inputs and the unit choice still affect what the number means in practice.
Accuracy of R, r, and H
A small error in any of the three inputs is amplified because the formula squares R and r and the slant height depends on (R - r). A 1 percent error in a radius becomes about a 2 percent error in its circle area, and the slant height is sensitive when R and r are close.
Perpendicular vs slanted height
H must be the perpendicular distance between the two base planes, not the side-wall length. Using the slant height as H gives a wrong slant height in the result panel and inflates the total area.
Whether the top is open or closed
The total area formula assumes both base circles are present. If the top is open, subtract pi * r^2 from the total to get the outside surface area. If the inside is also coated, the lateral area and inner base may need to be added separately.
Rounding when the value is reused
The result panel shows four decimal places, which is plenty for most quoting. For downstream work, keep at least four decimals to avoid drift and recompute the area at the end from the unrounded inputs.
- • The calculator is for right circular conical frustums. An oblique frustum, where the apex is not over the center of the bases, has a different lateral area formula and is not the same shape as the truncated cone on this page.
- • The page does not draw the frustum, accept an angle in place of one of the radii, or compute the centroid. Floating point rounding means the last digit of the total area can differ from a hand calculation that uses a truncated value of pi.
According to Omni Calculator, a frustum with R=8, r=2, and slant height S=12 has a total surface area of pi * (8^2 + 2^2 + 12 * 10) = 591.
If the closure is a domed cap rather than a flat circle, the sphere volume calculator covers the curved end pieces that the flat pi * r^2 term does not include.
Frequently Asked Questions
Q: What is the formula for the area of a frustum of a cone?
A: The total surface area is A = pi * (R^2 + r^2 + S * (R + r)), where R is the larger base radius, r is the top radius, and S is the slant height. The slant height is S = sqrt(H^2 + (R - r)^2) from the right triangle formed by the height and the difference of the radii.
Q: How do I find the slant height of a frustum of a cone?
A: The slant height is S = sqrt(H^2 + (R - r)^2). Square the perpendicular height, square the difference of the two radii, add the two squares, and take the square root. For a 30 cm tall frustum with R = 25 cm and r = 18 cm, S = sqrt(900 + 49) = sqrt(949) ≈ 30.81 cm.
Q: What is the lateral surface area of a frustum of a cone?
A: The lateral area is the slanted wall only, A_side = pi * S * (R + r). For R = 5 cm, r = 1 cm, and S = 10.77 cm, the lateral area is pi * 10.77 * 6 ≈ 203.02 cm^2, not counting the two base circles.
Q: Does the area formula work when the top radius is zero?
A: Yes. When r = 0, the top circle area becomes 0, the slant height becomes sqrt(H^2 + R^2), and the total area reduces to the surface area of a complete cone. The same closed-form formula gives the cone answer without a separate calculation.
Q: What is the difference between the lateral area and the total area of a frustum?
A: Lateral area is the slanted wall only, pi * S * (R + r). Total area is lateral area plus both base circles, pi * (R^2 + r^2 + S * (R + r)). For an open-top frustum, use the lateral area plus only the larger base.
Q: How do I convert a frustum area to gallons or liters for a planter or bucket?
A: Area tells you how much material covers the surface, not how much it holds. To find the volume, use the truncated cone volume formula V = (pi * H / 3) * (R^2 + R * r + r^2) and then convert with the same length unit to gallons or liters in a volume converter.