The magnetic flux \(\emptyset_{ B }\) through the surface or loop of vector area \(A\) is the magnetic field or number of field lines passing through the surface. It is defined as:
\(\emptyset_{ H }=B \cdot A,=B A \cos \theta\)
Where:
\(B=\) magnitude of the magnetic field,
\(A=\) area of the surface/loop \(\left( m ^{2}\right)\)
\(\theta=\) angle between the magnetic field lines and the normal or perpendicular to the surface.
where \(n=\) normal to the surface loop with area \(A\).
Key note:
The key idea to the problem at hand is to note that since the loop is placed in the \(x-y\) plane, the normal \(n\) to the surface will be in the \(z\) - direction. Also, it is stated in the problem that the magnetic field is pointing in the positive \(z\) - direction, thus, the normal \(n\) to the surface and the magnetic field are in the same direction. So \(\theta=0\). See the figure belowinally, using
$$
\begin{aligned}
\Phi_{ B } &=B a \cos \theta \\ &=\pi r^{2} \\
\Phi_{B} &=0.36 \times \frac{22}{7} \times 0.25^{2} \\
\Phi_{B} &=\frac{0.495}{7}=0.07 wb
\end{aligned}
$$