As a kid, looking at a hexagon-shaped jungle gym on the school playground, I remember wondering, just out of curiosity, what all those pointy corner angles would add up to. The formula I learned later was just one line, (n-2)×180° — I just memorized it at first, but once I later understood why it comes out that way, it became a lot more interesting.
The key is splitting it into triangles. Draw diagonals from one vertex of any polygon to all the other vertices, and the shape splits into several triangles. A pentagon, for example, splits into three triangles with two diagonals. One triangle's interior angles always sum to 180 degrees, so three triangles give 180×3=540 degrees as the pentagon's interior angle sum. Every extra side adds one more triangle, which is exactly where the formula (number of sides-2)×180 comes from.
A common point of confusion for students learning this for the first time is "why does it have to be from just one vertex?" Actually, it doesn't matter where you draw the diagonals from, but drawing them all from one vertex makes counting the triangles cleanest and least confusing, which is why it's taught that way. Draw the diagonals yourself with a ruler on a pentagon, hexagon, and heptagon, and the idea clicks quickly.
This concept turns out to come up surprisingly often in everyday life. Why a honeycomb's hexagonal cells are so stable, and why a soccer ball's pentagon and hexagon pieces mix in exactly that ratio, both come back to the interior angle sum. Calculating a polygon's angles is also fundamental when designing roof trusses in architecture.
One advanced question that often comes up in this unit is "what's one interior angle of a regular polygon?" Once you've found the interior angle sum, just divide it by the number of sides — for instance, a regular hexagon has an interior angle sum of 720 degrees, so one interior angle is 720÷6=120 degrees. It's also worth knowing, on the flip side, that the sum of the exterior angles (the outside angle formed by extending one side) is always 360 degrees, no matter the polygon's shape. It's like someone walking once around a polygon's perimeter, turning at every vertex — by the time they've made it all the way around, they've turned exactly 360 degrees back to their original direction. Connecting this exterior-angle-sum idea into the explanation lets kids grasp a much bigger picture: "oh, no matter how big or small the polygon, going all the way around always works out the same."
When studying this with your child, rather than having them memorize the formula first, I'd recommend cutting a polygon out of paper, actually drawing the diagonals, and counting the triangles by hand. Dividing it up with your own hands makes a formula that was just numbers on a page stick in memory much longer. On our activity page, you can change the number of sides with a slider and watch the triangles split automatically, so you can get a similar effect without paper and scissors. Start with triangles and quadrilaterals and gradually increase the number of sides through pentagons and hexagons, letting your child discover the pattern themselves, and you'll get to witness the moment they say, before you've even told them the formula, "wait, it grows by 180 degrees every time a side gets added?" That moment is the surest sign that they've truly understood the formula.