Cross Sectional Area

Calculating Cross-sectional Area

What is cross-sectional area?

The cross-sectional area refers to the area of a particular section or slice of an object. It is the area you get if you "cut" the object at a certain point, like slicing a loaf of bread. This cut provides a 2D view of the object's interior at that point.

Internal cross-sectional area

If you cut a cylindrical pipe across its width, the cross-sectional area is the area of the circle you see in the cut. When it comes to working out how much energy can move through a pipe, we need to know the cross-sectional area of the inside of the pipe, shown in Blue.

Calculating Cross-sectional Area

How to calculate cross-sectional area?

Internal

Diameter

When it comes to pipe for heating systems, UK pipe sizes refer to the total diameter of the pipe. For our calculations we need to know the internal diameter as this is the space occupied by the water flowing through it.

Internal

There are two equations for working out the cross-sectional area of a circle, it's really just a case of preference as to which one you use.

Radius

The most commonly used is is 𝐴=𝜋𝑟2

The other method is A=𝜋(d2/4)

Calculating Cross-sectional Area

Worked example for 22mm pipe:

Total

For this example we are going to look at 22mm copper pipe.

Diameter

We know that 22mm tells us the total diameter of the pipe, so the first thing we need to do is find out the thickness of the pipe walls so we can deduct these from our total diameter to get our internal diameter.

22mm

This information should be available from the pipe manufacturer as there are different grades of copper pipe, which have different wall thicknesses depending on their intended application.

1mm

For this example we will use Type M, which is light duty pipework with the thinnest walls and is commonly used in domestic plumbing. Type M 22mm copper pipe typically has a wall thickness of 1mm, depending on manufacturer.

Wall Thickness

Calculating Cross-sectional Area

Worked example for 22mm pipe:

Wall #1

Now we know that our 22mm copper pipe has a wall thickness of 1mm, we need to double this as there are two walls that make up the total diameter.

Wall #2

1mmx2=2mm Total wall thickness.

Internal

Diameter

20mm

Now we have the total wall thickness, we deduct this from the total diameter to get our internal diameter: 22mm-2=20mm.

Once we have our internal diameter we can half this to give us the radius: 20/2=10mm.

Calculating Cross-sectional Area

Worked example for 22mm pipe:

Now we know the internal diameter and radius, there's one last step before we carry out either of the equations.

We need our final answer to be in m2, so we need to divide our figures by 1000, otherwise we have to the the conversion after everything has been squared.

Area needs to be m2

So the internal diameter is 20mm/1000=0.02m

The internal radius is 10mm/1000=0.01m

Calculating Cross-sectional Area

Worked example for 22mm pipe:

If using a scientific calculator, you can simply press the button marked 𝜋 which is the symbol for Pi

If you don't have this function, it's common for Pi to be rounded to 3.14, which is what we will use for the examples shown below:

0.000314 m2

Method 1) 𝐴=𝜋(𝑟2) So: 0.01x0.01=0.0001x3.14=0.000314m2

Method 2) A=𝜋(d2/4) So: 0.02x0.02=0.0004/4=0.0001x3.14=0.000314m2

Calculating water volume of pipe

Worked example for 22mm pipe:

One application for cross-sectional area is for calculating the volume of water a given length of pipe can carry.

Volume for a pipe (V) = internal cross-sectional area (A) x Length (L)

Continuing with our example of 22mm copper pipe with a cross-sectional area of 0.000314m2, we then multiply this by the length of pipe. For this example we will use 10m.

0.000314 m2

So: 0.000314 x 10 = 0.00314m3 Next, we want to convert this volume in m3 to litres. There's 1000 litres in 1 cubic metre, so we need to multiply by 1000. So: 0.00314 x 1000 = 3.14 litres in 10 metres of 22mm copper pipe