Area

Prof. Dr. Thorsten Jungmann (last updated 2025-09-03)

Area (German: Fläche) is a general mathematical quantity. In electrical engineering, areas occur, for example, in the context of conductor cross-sections.

1 Quantity symbol and unit

The quantity symbol of area is $A$. Its dimension (unit) is square metre, abbreviated with the unit symbol ${\mathrm{m}}^2$.

$$[A]={\mathrm{m}}^2$$

(read: ‘The dimension of A is square metres.’)

2 Practical and worked examples

The cross-section becomes visible when you look at a component perpendicular to its longitudinal direction or ‘cut through’ it.

• For a round conductor (wire), the cross-section is a circle.
• For a stranded wire, the cross-section consists of many small circular areas of the individual wires.
• For a strip or bar conductor, the cross-section is rectangular.
• For capacitor plates, the cross-section corresponds to the effective area of the plates, which we assume to be a rectangle.

2.1 Conductor cross-section

The cross-sectional area describes the area of a body perpendicular to its longitudinal axis or to the direction of current flow and largely determines the electrical properties of conductors and other components such as capacitors.

For circular cross-sections: $A=\pi \cdot r^2=\frac{\pi \cdot d^2}{4}$ with the radius $r$ or diameter $d$.

In practice, the cross-sectional area of conductors is often stated in square millimetres: $[A]=\mathrm{m}{\mathrm{m}}^2$, where $1\mathrm{m}{\mathrm{m}}^2=10^{-6}{\mathrm{m}}^2$.

The cross-section largely determines the electrical properties of a conductor/electric line/cable:

• The larger the cross-section, the smaller the resistance and the larger the current that can be carried safely.
• The smaller the cross-section, the higher the current density for the same current.

Examples:

• Resistance of a copper wire
• Current density in a conductor

2.2 Area of capacitor plates

For rectangular cross-sections: $A=a\cdot b$.

The cross-section largely determines the electrical properties of the capacitor:

• The capacitance increases with a larger plate area (and thus a larger cross-section).
• The parallel connection of two technically identical capacitors corresponds to doubling the plate area.

Examples:

• Capacitance of a capacitor
• Magnetic flux density (?) in an inductor

3 Typical orders of magnitude

While bond wires in integrated circuits have cross-sections of a few $\mu {\mathrm{m}}^2$, installation cables typically have $1.5$ to $16\mathrm{m}{\mathrm{m}}^2$. High-voltage cables can reach cross-sections of more than $1000\mathrm{m}{\mathrm{m}}^2$. Busbars in substations have cross-sections of several $\mathrm{c}{\mathrm{m}}^2$.

Copper electric line in the domestic installation

The cross-section of a typical domestic installation cable protected by a 16 A fuse:

$$A=1.5\mathrm{m}{\mathrm{m}}^2=1.5\cdot 10^{-6}{\mathrm{m}}^2$$

4 Example phrasings

• ‘The electric line has a cross-section of $6\mathrm{m}{\mathrm{m}}^2$.’
• ‘For this current, a minimum cross-section of $2.5\mathrm{m}{\mathrm{m}}^2$ is required.’
• ‘The conductor cross-section was increased from $4\mathrm{m}{\mathrm{m}}^2$ to $10\mathrm{m}{\mathrm{m}}^2$.’
• ‘With a cross-section of $1.5\mathrm{m}{\mathrm{m}}^2$, the continuous current must not exceed $16\mathrm{A}$.’

5 Related quantities

• The cross-section determines the electrical resistance of a conductor:

$$R=\rho \cdot \frac{l}{A}$$

If you double the cross-section, the resistance is halved.

• The current density is the quotient of current and cross-section:

$$J=\frac{I}{A}$$

• The capacitance of a parallel-plate capacitor depends largely on the plate area:

$$C=\epsilon \cdot \frac{A}{d}$$

• The maximum permissible current-carrying capacity of a conductor is largely determined by its cross-section:

$$I_{\max }=J_{\mathrm{rated}}\cdot A$$

6 Relevant components

• Conductors, electric lines and cables: dimensioning according to current-carrying capacity
• Transformers: calculation of the winding cross-sections
• Capacitors: plate area largely determines capacitance