How to Find the Height of a Cone
h = 3V / πr² or h = √(l² − r²)
From the volume, multiply by 3 and divide by the base area. From the slant, use Pythagoras. A cone of radius 3 holding 37.70 has a height of 4, and so does a cone of radius 3 with a slant of 5.
Finding the height of a cone is really two questions that share a name, and the first job is working out which one you have been asked. If you know how much the cone holds, the volume formula runs backwards in one step. If you have measured down the outside of it, you have the slant, and the slant is a different length that needs Pythagoras before it becomes a height.
The slant and the height are not the same line
Stand a cone on a table. The height is the straight drop from the tip to the middle of the base, hidden inside the solid. The slant is the distance from the tip down the outside to the rim, and it is always longer, because it is the hypotenuse of the right triangle that the height and the radius make between them.
That triangle is the whole lesson. Radius, height and slant obey r² + h² = l², so any two of them give you the third. The worked example uses 3, 4 and 5 because it is the friendliest right triangle there is, and it makes every check in this article come out exact.
Check what you have
A volume, a slant, a surface area or an angle. Each one reaches the height by a different route, and they are not interchangeable.
Get back to r and l
Every route ends up at the same right triangle. Surface areas and angles are just slower ways of arriving at the slant.
Finish the triangle
h = √(l² − r²), or straight from the volume with 3V over πr². For the worked cone both give 4.
A worked check you can repeat on paper
Six routes, one height. Notice one oddity in that table: the lateral surface reads 47.12 and so does the volume you get by wrongly using the slant as the height. Those two numbers agreeing is a coincidence of this particular cone, where the radius happens to be 3, and it means nothing. It is worth pointing out precisely because a repeated number is exactly the sort of thing that convinces people they have found a shortcut.
When you have a surface area instead
This case sounds harder than it is, as long as you know the radius. The curved side of a cone is πrl, so dividing the lateral surface by πr hands you the slant directly, and from there it is Pythagoras again. A lateral surface of 47.12 on a radius of 3 gives a slant of 5, so the height is 4.
Total surface area needs one extra move, because it includes the base circle. Subtract πr² first. A total of 75.40 on a radius of 3 has a base of 28.27, leaving a curved side of 47.12, which is the case above. What makes this easy is that the radius was given. If the radius is the unknown as well, the radius appears both squared and on its own, and you have a quadratic rather than a rearrangement.
Measuring a stockpile with nothing but a tape
Nobody climbs a pile of gravel to measure its height, and you cannot get a level from the tip to the ground without standing where the pile is. What you can do is walk the base and run a tape up one side, which gives you a diameter and a slant. That is enough.
Take a pile 12 m across with a slant of 7.5 m. The radius is 6, so the height is √(56.25 − 36), which is √20.25, which is 4.5 m. The volume is a third of π times 36 times 4.5, or 169.65 cubic metres. That pile also sits at 36.87 degrees, which is a believable angle of repose for crushed stone and a useful sanity check in itself: dry sand tends to settle nearer 34 degrees, so a measured angle far outside the low thirties usually means the pile is not a clean cone or the tape wandered.
| What you know | Route to the height | Worked value |
|---|---|---|
| Volume and radius | 3V divided by πr² | 4 |
| Slant and radius | square root of l² minus r² | 4 |
| Lateral surface and radius | l = A over πr, then Pythagoras | 4 |
| Total surface and radius | subtract πr², then as above | 4 |
| Base angle and radius | r times the tangent of the angle | 4 |
| Diameter and slant, on a stockpile | square root of l² minus r², in metres | 4.5 |
Mistakes that catch people out
- Using the slant as the height. The one that costs the most. On the worked cone it turns 37.70 into 47.12, exactly 25 percent too much, and the wrong answer looks entirely reasonable because it is the right sort of size.
- Forgetting the 3 when working backwards. The volume has a third in it, so undoing it means multiplying by 3, not dividing. Miss it and your height comes out three times too small.
- Using the diameter as the radius. A pile 12 m across has a radius of 6, not 12. In the Pythagoras step that error goes inside a square, and with a slant of 7.5 it would leave you taking the square root of a negative number, which is at least an honest error message.
- Subtracting the base twice, or not at all. Lateral surface is the curved side only. Total surface includes the base circle. Check which one you were handed before dividing by πr.
- Treating a leaning cone as an upright one. If the tip is not over the centre of the base, the height is still the perpendicular drop to the base plane, and r² + h² = l² no longer holds for the slant you can measure. The volume formula survives, the triangle does not.
Try these
- A cone holds 100 cubic units and has a radius of 5. Find its height to two decimal places.
- A cone has a radius of 5 and a slant of 13. Find its height.
- A cone has a radius of 5 and a height of 12. Check your answer to question 2 by finding its volume.
- A cone has a radius of 5 and a lateral surface area of 204.20. Find its height.
- Double the height of the worked cone and leave the radius alone. What happens to the volume, and what happens to the slant?
Show the answers
- 3 times 100 is 300, and π times 25 is 78.54, so the height is 300 over 78.54, which is 3.82.
- √(169 − 25) is √144, which is 12. Another Pythagorean triple, 5, 12 and 13.
- A third of π times 25 times 12 is 314.16 cubic units.
- Divide by πr: 204.20 over 15.71 gives a slant of 13, so the height is 12 again. Same cone, different starting information.
- The volume doubles, from 37.70 to 75.40, because height appears once and only once. The slant does not double: it goes from 5 to √(9 + 64), which is 8.54.
Common questions
How do you find the height of a cone from its volume?
Multiply the volume by 3 and divide by πr². A cone of radius 3 holding 37.70 gives 113.10 over 28.27, which is a height of 4. The 3 is there because the volume formula has a third in it.
What is the difference between the height and the slant height?
The height is the perpendicular drop from the tip to the centre of the base. The slant runs down the outside to the rim and is always longer. They are the two shorter sides and the hypotenuse of the same right triangle, so r² + h² = l².
How do you find the height from the slant height?
Take the square root of the slant squared minus the radius squared. A radius of 3 with a slant of 5 gives √(25 − 9), which is 4. A radius of 5 with a slant of 13 gives 12.
Can you find the height from the surface area?
Yes, provided you know the radius. Divide the lateral surface by πr to get the slant, then use Pythagoras. If you were given a total surface area, subtract the base circle πr² first.
What happens if I use the slant height in the volume formula?
You overstate the volume, and by more than people expect. On a cone of radius 3, using a slant of 5 where the height of 4 belongs turns 37.70 into 47.12, which is exactly 25 percent too much.
How do you measure the height of a conical stockpile?
Do not measure the height at all. Walk the base for the diameter and run a tape up one side for the slant, then let Pythagoras do the rest. A pile 12 m across with a slant of 7.5 m is 4.5 m tall and holds 169.65 cubic metres.
Related lessons and calculators
- Volume of a cone, the formula this lesson runs backwards, and the place the one third comes from.
- Surface area of a cone, where πrl is unrolled into a flat sector and the slant earns its keep.
- Missing side of a right triangle, the Pythagoras step on its own, for when the triangle is the part giving trouble.
- Radius of a cylinder, the same kind of rearrangement on the solid a cone is a third of.
- Volume of a cylinder, useful as a check, since a cone always holds a third of the cylinder around it.
- Volume of 3D shapes calculator, including the stockpile and frustum cases for piles that are cut off at the top.
- Area formulas for every common shape, the index this lesson sits inside.
- Wolfram MathWorld on the cone, for the general treatment including oblique cones.
- Height of a cylinder, the same question without a slant to trip over.