Question about arc dividing


Maybe someone can help me. Is it possible to measure the arc to a certain point. I would like to go on with 240mm steps. How can I measure 240mm on arc?
Hope you understand what I mean. Photo below.

Thank you.

Do you mean that if the arc had a length of, say, 960mm you would want to measure a quarter of its total length?

It would help to know why you want to do this and what you mean by “measure”. Do you want to find the point at that length or do you want to place a dimension? By length, do you mean along the arc itself or the notional straight line along the base of a segment?

I am constructing a L-shaped stair. And this is the top view. All the steps should have a even 240mm depth from the front to the back. For the straight steps I take the measure from the middle of the step to the end and these are all 240mm. Now when I arrive to the point where I have to make corner steps, then these should also have 240mm depth from the middle point If I measure it by the arc.

I would like to find the point along the arc where the step is 240mm wide.

Thank you.

Like this?

I have assumed that what you mean by corner steps is what are normally called winders or kite shaped treads. In theory, you can have any number of those but in practice it is usually between 3 and 5 with 3 being commonest. If you have a specific dimension to achieve at the centre, that will dictate the width of your stair at that point. In your example, the dimension from the meeting point of the treads to the arc will be 464mm. This can be calculated by trigonometry. If you wanted the width of the carriageway to be 800mm (say), it would mean that the meeting point of the treads would be at a notional point 64mm outside the carriageway. This might fall somewhere within the newel post.

I produced that drawing by geometry but I could have started with trig just as easily.

one way…


1 Like

You can use TIGs “Arc by Tool” plugin to draw an arc with a specific radius and length…

“240 from the middle” isn’t practicle. It may depend on the width of the stairs. See example in the image. I would play with the option to ‘Divide’ the arc (right click context menu) to find / narrow down the location and the number of steps needed to overcome the 90 degrees.

And as you can see, 4 steps instead of 3 also affects the rise per step, one extra compared to 3 steps. You need to take other things into account too.


what about using some maths ?

if you want to find the angle that will give you the 240 mm arc length you can use this formula :

angle = (240x180) / (radius x π)

you then just have to rotate a copy of your step line with the given angle

Or another way…
The 240mm [as the central-tread min dim] is the short edge of an isosceles triangle with sides of 500mm [if that’s the mid-line of the tread].
Drawing in a central Line from the apex to the middle of the 240mm gives you a right-angled triangle with Hypotenuse =500, Opposite=120 and its Adjacent side unknown.
Good old Pythagoras…
Ad = sqrt((500*500) - (120*120)) = 485.396444..
So draw a 90° triangle with sides that and 120mm and the Hypotenuse will measure 500mm for all practical purposes…
You are building a set of steps, not a jet-engine - so if your builder is within 1mm it’ll be a miracle !

Hold on, why isn’t the base line distance 464mm?

So just for the record, this is how I would approach it using trigonometry.

If there are to be 3 winders to a 90 degree turn, each nosing is at 30 degrees to one another.

If the distance across the base of the isosceles triangle with two sides described by nosings is to be 240mm, we have to bisect the triangle to create two right angled triangles back to back, each with an angle of 15 degrees at the apex and with base of 120mm.

The hypotenuse is obtained by the formula 120mm ÷ sin 15 degrees = 464mm. So the centreline of the treads has to be that distance from the meeting point of the winders.

Or do I need a refresher course on trig?

it’s actually 463.644397 which rounds up to 464 when viewed without decimal places…