A cone is fixed by two measurements: the radius of its round base and either its perpendicular height or its slant height. Give this calculator any two of the three and it works out the third, then the volume, the base area, the curved (lateral) surface area, the total surface area, the apex angle, and the flat pattern you would cut to make the cone from sheet material.
How to use this cone calculator
- Choose what you know. Pick “Radius and perpendicular height” if you measured straight up from the centre of the base to the tip. Pick “Radius and slant height” if you measured along the sloping side, which is what a tape laid on the surface of a real cone gives you.
- Enter the radius. The radius is half the width across the base. If you measured the full width (the diameter), halve it before typing it in.
- Enter the second measurement in the same unit as the radius, choose that unit from the list, and press Calculate.
- Read the full set of results: the missing height, volume, three areas, apex angle, a labelled cross-section diagram, and the flat pattern figures for cutting the curved surface from flat material.
The formulas, and perpendicular versus slant height
The two heights are the part of cone maths that trips people up, so they are worth separating clearly. The perpendicular height (h) runs straight up inside the cone, from the centre of the base to the apex, at a right angle to the base. The slant height (l) runs along the outside surface from the rim to the apex. The radius, the perpendicular height and the slant height always form a right-angled triangle, so Pythagoras connects them:
With any two of the three known, the third follows, and every other result is built from r, h and l:
The 3 in the volume formula is exact, not an approximation: a cone holds exactly one third of the cylinder that shares its base and height. The lateral area formula uses the slant height, never the perpendicular height, because the curved surface is measured along the slope.
The flat pattern
Unroll the curved surface of a cone and it becomes a sector of a circle: a wedge with a curved edge. The wedge has radius equal to the slant height l, and its curved edge must wrap exactly around the base, so its arc length is the base circumference 2πr. The sector angle follows from that:
The apex half-angle, the angle between the cone’s centre line and its side, is atan(r / h); the full apex angle is twice that. A narrow, pointed cone has a small apex angle and a small sector angle relative to its slant; a wide, flat cone pushes the sector angle towards 360°.
Worked examples
Example 1: the 3-4-5 cone (height known)
A cone has radius 3 in and perpendicular height 4 in. The slant height is √(3² + 4²) = √25 = 5.00 in, the classic 3-4-5 triangle. Volume = π × 9 × 4 / 3 = 12π = 37.70 cu in. Base area = 9π = 28.27 sq in. Lateral area = π × 3 × 5 = 15π = 47.12 sq in. Total surface area = 24π = 75.40 sq in. The apex half-angle is atan(3 / 4) = 36.87°, and the flat pattern is a sector of radius 5.00 in with angle (3 / 5) × 360 = 216.00°.
Example 2: recovering the height from the slant (slant known)
A paper cone has radius 5 cm and the sloping side measures 13 cm. The perpendicular height is √(13² − 5²) = √144 = 12.00 cm. Volume = π × 25 × 12 / 3 = 100π = 314.16 cu cm. Lateral area = π × 5 × 13 = 65π = 204.20 sq cm, base area = 25π = 78.54 sq cm, total = 90π = 282.74 sq cm. To make this cone, cut a sector of radius 13.00 cm and angle (5 / 13) × 360 = 138.46°; its curved edge is 2π × 5 = 31.42 cm long, exactly the circumference of the base.
Example 3: starting from a diameter
A funnel-shaped cone is 12 in across the top and 9 in deep. Halve the width first: radius 6 in, height 9 in. Slant height = √(36 + 81) = √117 = 10.82 in. Volume = π × 36 × 9 / 3 = 108π = 339.29 cu in. Lateral area = π × 6 × 10.82 = 203.89 sq in, and the total surface area including the base is 316.99 sq in. Typing the diameter in as the radius is the single most common cone error, and it would have quadrupled the volume here.
Reading your results
Volume is the space inside the cone, in cubic units. It grows with the square of the radius, so doubling the radius quadruples the volume, while doubling the height only doubles it.
Lateral area is the curved surface only. That is the figure you need for covering, wrapping, painting the outside of, or cutting material for an open cone such as a lampshade, a hopper or a paper cone, because none of those have a base disc.
Total surface area adds the base disc to the lateral area. Use it for a solid cone, or a closed container, where the base exists and needs material too. If your cone is open at the bottom, the total overstates what you need; use the lateral area.
Slant height and perpendicular height are both printed whichever mode you used, so you can check a measurement taken the other way: measure along the slope on the real object and compare it with the slant figure here.
Flat pattern figures give the sector radius, the sector angle and the arc length. Draw the sector on your material with a compass set to the sector radius, mark the angle, cut it out, and roll it so the two straight edges meet. Add a small overlap tab of your own if you plan to glue the seam; the pattern here is the exact cone surface with no allowance added.
Common mistakes
- Using the diameter as the radius. Every formula needs the radius. A diameter typed into the radius box multiplies the volume by 4 and every area by 4.
- Putting the perpendicular height into the area formula. Lateral and total area are built on the slant height. With the 3-4-5 cone, using 4 instead of 5 gives a lateral area of 37.70 sq in instead of the correct 47.12 sq in.
- Forgetting the divide by 3. πr²h on its own is the volume of the matching cylinder, three times the cone.
- Mixing units. A radius in centimetres and a height in inches produces numbers that look plausible and are simply wrong. Convert both to one unit first.
- Measuring the slant with a straight ruler held off the surface. The slant is the surface distance from rim to apex; on a real cone, lay the tape on the surface.
Limitations
- This calculator covers right circular cones: a round base with the apex directly above its centre. Oblique cones, whose apex sits off-centre, need different mathematics and are not covered.
- A truncated cone (frustum), with the tip cut off, has two radii and is a different shape with different formulas; it is not calculated here.
- Results assume a perfect geometric cone. Real objects have wall thickness, seams and rounded tips, so treat the figures as exact for the shape you typed in and as close estimates for a manufactured object.
- All results are shown rounded to two decimal places; the underlying calculation keeps full precision.
Frequently asked questions
What is the difference between slant height and perpendicular height?
The perpendicular height is the straight internal distance from the centre of the base up to the apex. The slant height is the distance along the sloping surface from the rim to the apex. The slant height is always the longer of the two, and the two are connected by Pythagoras: l² = r² + h².
I only know the slant height. Can I still get the volume?
Yes. Choose “Radius and slant height” above. The calculator first recovers the perpendicular height with h = √(l² − r²), then computes the volume and areas as usual. The slant height must be longer than the radius; if it is not, no real cone matches those numbers and the calculator says so.
How do I make a cone out of paper or sheet metal?
Cut a sector of a circle: radius equal to the slant height, angle equal to the sector angle in your results. Roll it until the two straight edges meet and the curved edge forms the base circle. The calculator prints both figures, plus the arc length, which you can check against the base circumference before cutting.
Does the total surface area include the base?
Yes. Total surface area is the lateral (curved) area plus the base disc. For an open cone with no base, use the lateral area figure on its own.
Why is the volume exactly one third of a cylinder?
It is a proven result of geometry, not a rule of thumb: any cone holds one third of the cylinder with the same base radius and the same height. You can test it with rice or sand and two matching shapes; the cylinder fills the cone three times.
Can this calculator do a cone with the top cut off?
No. That shape is a frustum and needs both the bottom and top radii. This page covers complete cones only, so every result matches one defined shape and is not stretched to cover a different one.
Related calculators
- Math calculators
- Cylinder Calculator: the matching straight-sided shape, useful for the one-third volume comparison.
- Sphere Calculator: volume and surface area of a ball from its radius.
- Pyramid and Prism Calculator: the flat-sided relatives of the cone.
Method: standard right-circular-cone geometry. V = πr²h/3; l = √(r² + h²); lateral area = πrl; total surface area = πr(r + l); sector angle = (r/l) × 360°. Figures checked against independently calculated fixtures. Last reviewed: 20 October 2026.
Disclaimer: This calculator gives estimates for general information only. It is not financial, medical or other professional advice.