How to determine the depth of a term in a custom taxonomy?

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I’m building a <select> dropdown to display a list of terms for a custom hierarchical taxonomy. How do I know the depth of each term? I’d like to add some indentation for child-terms (the more deeply they’re nested, the more indentation there should be).

However I’m not able to retrieve the depth level from get_terms(). How to get the depth level as an integer?

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  2. <?php
$args = array(
    'taxonomy' => 'category',
    'orderby' => 'name',
    'order' => 'ASC',
    'hierarchical'  => true,
    'hide_empty' => false,
);
$the_query = new WP_Term_Query($args);
$categories = $the_query->get_terms();


foreach($categories as $cat){
    $ancestors = get_ancestors( $cat->term_id, 'category' );
    $cat->ancestors = $ancestors; // array( 0 => 15, 1 => 45 ) - 3rd level term
    $cat->depth = count( $ancestors ) ;
}
?>
<select>
    <?php 
        foreach($categories as $cat){
            echo '<option value="'.$cat->term_id.'">'.str_repeat('&nbsp;&nbsp;&nbsp;',$cat->depth).$cat->name.'</option>';
        } 
    ?>
</select>