I feel weaved elements can add texture and interest to your layout. To create the circle you see here I cut 1/2 inch strips of two different papers, one patterned paper and one solid paper. I then took a very thin piece of 12 x 12 white cardstock and used it as a base to hold my strips in place while I worked on weaving the strips together. I used a removable adhesive that comes in a runner dispenser and placed adhesive along one edge of my cardstock. I laid all my solid color strips side by side securing one end of each strip in the adhesive. I gave myself just a tiny bit of space between the strips as I laid them out since the paper weaving process does need some room in order for the end product to lay flat.
After the solid color strips were secure, I used my removable adhesive to add additional adhesive to the right and left edges of my cardstock. With the adhesive on the edges of the cardstock, each strip I weave can be anchored on both sides and your weaving won't come apart as you work not matter how much you move the paper around.
Keep working up your page anchoring each strip as you go. When you have finished weaving the entire page you are ready to trace your circle on the woven page. The lines you draw for your circle will become your cutting lines.
When your circle (or any other shape) is drawn on the page, you can keep all these woven pieces together by stitching with a sewing machine just inside your cutting lines.
When your stitching is complete you can cut out your shape. Once your shape is cut you no longer need the base paper, so just carefully tear it away from the back of the circle and this will make your piece less bulky.
Here is my second example for you. In this layout I used on of Pam's templates as my background. It's the beautiful Die cut flower template. I then decided to make two individual woven flowers for the page. In this case my strips are just 1/4 inch wide because the shape is so much smaller.