You can achieve that by Binding the Height Property to the ActualWidth Property of the Grid, you can even go further, for example if you want to set the height to be equals to two times of the width, by using a converter :
Code example 1 :
<Grid x:Name="ContentPanel" Grid.Row="1">
<Button x:Name="Button" Content="Button" Width="200" Height="{Binding ElementName=ContentPanel, Path=ActualWidth}"/>
</Grid>
Code example 2 :
<Grid x:Name="ContentPanel" Grid.Row="1">
<Button x:Name="Button" Content="Button" Width="200" Height="{Binding ElementName=ContentPanel, Path=ActualWidth, Converter={StaticResource WidthToHeightConverter}}"/>
</Grid>