Create Polygonal View Frustum

I'm getting really close to being able to do real view frustum culling with triangle shaped or quad shaped polygons but need only one measly step to getting there.
Basically when I wanted to add frustum culling I did a number of techniques. First I tried testing the vertices of a poly to see if they are all in or out of the view frustum. Works great! Except one problem. When a polygon is big or you are close to it, all the vertices are out of the view frustum but the polygon remains to be seen. So I tried testing to see if the sides of the polygons were in the view frustum by using a line plane collision algorithm as a third and forth case, which is if the lines of the polygon collide into any of the 6 planes of the view frustum, it will be visible, else, cull the polygon. I was basically using this:

		for (int i = 0; i < 6; i++)
		{
			//Check if any of the 4 lines of the quad cross any of the 6 view frustum planes. If the values fall between 0 and 1, then the line has crossed
			//the plane and the polygon is visible.
			t[0] = -((frustum_plane[i].a * vertex[0].x + frustum_plane[i].b * vertex[0].y + frustum_plane[i].c * vertex[0].z) + frustum_plane[i].d) /
				     (frustum_plane[i].a * vector[0].x + frustum_plane[i].b * vector[0].y + frustum_plane[i].c * vector[0].z);
			t[1] = -((frustum_plane[i].a * vertex[1].x + frustum_plane[i].b * vertex[1].y + frustum_plane[i].c * vertex[1].z) + frustum_plane[i].d) /
				     (frustum_plane[i].a * vector[1].x + frustum_plane[i].b * vector[1].y + frustum_plane[i].c * vector[1].z);
			t[2] = -((frustum_plane[i].a * vertex[2].x + frustum_plane[i].b * vertex[2].y + frustum_plane[i].c * vertex[2].z) + frustum_plane[i].d) /
				     (frustum_plane[i].a * vector[2].x + frustum_plane[i].b * vector[2].y + frustum_plane[i].c * vector[2].z);
			t[3] = -((frustum_plane[i].a * vertex[0].x + frustum_plane[i].b * vertex[0].y + frustum_plane[i].c * vertex[0].z) + frustum_plane[i].d) /
				     (frustum_plane[i].a * vector[3].x + frustum_plane[i].b * vector[3].y + frustum_plane[i].c * vector[3].z);

			if ((t[0] < 0.0f || t[0] > 1.0f) &&
				(t[1] < 0.0f || t[1] > 1.0f) &&
				(t[2] < 0.0f || t[2] > 1.0f) &&
				(t[3] < 0.0f || t[3] > 1.0f))
			{
				pierced[i] = false;
			}
			else
			{
				pierced[i] = true;
			}
		}

Where the vector was literally all 4 sides of a quad. Works but ran into another problem. Even when the polygon is completely off screen, it still manages to collide into a frustum plane due to the fact it is literally 6 infinite planes. So I wanna test the frustum as six polygons rather than treat them as planes. How can I make the view frustum 6 polygonal quads? Thanks in advance.