Special Figures in Geometry: Closed, Regular, and Symmetric Shapes

Imagine you’re playing a puzzle game where you need to trace a path without lifting your pencil, or creating a beautiful paper snowflake by carefully cutting folded paper. These activities aren’t just fun-they’re your first encounters with some very special figures in geometry. When children learn about shapes, they quickly discover that not all figures are created equal. Some have remarkable properties that make them stand out, and understanding these special characteristics helps young learners develop a deeper appreciation for the mathematical patterns all around them.

The mystery of closed and open figures 🔗

One of the first distinctions children encounter is whether a figure is closed or open. Think of it like this: if you were an ant walking along the outline of a shape, could you return to your starting point without ever leaving the line? If yes, you’ve found a closed figure.

A closed figure has the same starting and ending point. When you trace around it with your finger, you end up exactly where you began. Circles, squares, triangles, and rectangles are all closed figures. Open figures, on the other hand, have different starting and ending points. A curved line that looks like the letter C, a semicircle, or a line segment are examples of open figures.

Here’s something interesting: because open figures don’t close up completely, they don’t have a definite area. Imagine placing a pebble near a piece of string lying on the ground. Is the pebble inside or outside the string? You can’t really say, because the string doesn’t enclose a space. But if that string formed a complete circle, you could easily tell whether the pebble was inside or outside.

Teaching closed and open figures through games 🔗

Young students learn this concept best through hands-on exploration. Try the tracing game: give children various cutout shapes and ask them to trace the outline without lifting their pencil. If they can return to the starting point while staying on the line, they’ve discovered a closed figure. You can also play the rope game outdoors, where children form different shapes with a long rope and determine whether their creation is open or closed.

Another engaging activity involves sorting picture cards. Provide images of everyday objects-a door, a rainbow, a window, the letter O, the letter C-and have students categorize them as open or closed figures. This helps them recognize these mathematical concepts in the real world around them.

Distinguishing regular from irregular figures 🔗

Once children grasp the idea of closed figures, they’re ready to explore another special category: regular and irregular polygons. A regular polygon is like the perfectionist of the shape world-all its sides have exactly the same length, and all its angles are exactly the same size. An irregular polygon is more relaxed about these rules; its sides and angles can be different sizes.

Think of a square. Each of its four sides is identical in length, and each of its four corners forms a perfect right angle. That’s what makes it a regular quadrilateral. Now think of a rectangle that isn’t a square-it still has four sides and four right angles, but two sides are longer than the other two. This makes it an irregular quadrilateral.

The same principle applies to triangles. An equilateral triangle, with all three sides equal and all three angles equal, is a regular triangle. But an isosceles triangle (two equal sides) or a scalene triangle (no equal sides) are irregular triangles.

Hands-on activities for exploring regularity 🔗

One effective activity involves giving students sets of triangles cut from paper-some equilateral, some isosceles, some scalene. Ask them to measure the sides with a ruler and the angles with a protractor. They’ll discover for themselves which triangles are regular and which are irregular. This measurement experience makes the abstract concept concrete and memorable.

Paper-cutting activities work wonderfully too. Have students fold a piece of paper and cut out shapes. When they unfold their creation, they can examine whether the resulting shape is regular or irregular. Challenge them to create a regular hexagon by folding paper into sixths and making strategic cuts. This combines spatial reasoning with understanding of regular polygons.

For quadrilaterals, provide students with straws or craft sticks of various lengths. Ask them to create different four-sided shapes and determine whether each is regular or irregular. Can they make a square? A rectangle? A parallelogram? What makes each one special or different?

The beautiful world of symmetry 🔗

Perhaps the most visually striking property that makes some figures special is symmetry. A figure has symmetry when one half is a mirror image of the other half. The imaginary line that divides a symmetrical figure into two matching halves is called the line of symmetry, and discovering lines of symmetry is one of the most engaging activities for young mathematicians.

Symmetry surrounds us everywhere in nature. A butterfly’s wings, a leaf, a human face, a snowflake-all display symmetry. When children learn to recognize symmetry in geometric figures, they begin noticing it everywhere in their environment, from architecture to art to biology.

Paper folding to discover symmetry 🔗

The paper-folding method is one of the most powerful tools for teaching symmetry to elementary students. Here’s how it works: give a child a paper cutout of a shape-perhaps a square, rectangle, or triangle. Ask them to fold it in half so that one side lies exactly on top of the other. If the edges and corners match up perfectly, that fold line is a line of symmetry.

Some shapes have multiple lines of symmetry. A square has four: you can fold it in half vertically, horizontally, or along either diagonal, and it will match up perfectly each time. A rectangle has only two lines of symmetry: vertical and horizontal folds work, but diagonal folds don’t. An equilateral triangle has three lines of symmetry, one through each vertex. Meanwhile, a scalene triangle has no lines of symmetry at all.

Students can use highlighters to mark each line of symmetry they discover. This creates a visual record of their exploration and helps them compare different shapes. Which shapes have the most lines of symmetry? Which have none? These questions lead to fascinating mathematical discoveries.

Mirror activities for visualizing symmetry 🔗

Another captivating approach involves using small mirrors. Place a mirror perpendicular to a piece of paper and draw half of a simple picture-perhaps half of a butterfly or half of a heart. When children look in the mirror, they see the complete symmetrical image. They can then try to draw the other half to match what they saw in the mirror.

This mirror technique works beautifully with letters and numbers too. Which letters of the alphabet have vertical symmetry? (A, H, M, O, T, U, V, W, X, Y) Which have horizontal symmetry? (B, C, D, E, H, I, K, O, X) Some letters, like O and X, have both. This activity connects geometry to literacy in a meaningful way.

Creating symmetrical art 🔗

One of the most delightful symmetry activities combines art with mathematics. Have students fold a piece of paper in half, paint or draw on one side only, then fold and press the paper together while the paint is still wet. When they open it, they discover a perfectly symmetrical design. This paint-and-fold technique creates beautiful butterfly paintings, abstract designs, or even symmetrical monsters.

Paper snowflakes offer another wonderful symmetry experience. Students fold paper multiple times and make strategic cuts. When they unfold their creation, they discover intricate designs with multiple lines of symmetry. Each fold they make adds another axis of symmetry to the final design.

Finding special figures in everyday life 🔗

The real magic happens when children start recognizing these special figures in the world around them. A stop sign is a regular octagon. A honeycomb is made of regular hexagons. Many flowers display radial symmetry. Buildings often feature symmetrical facades. Traffic signs, furniture, tiles, and countless other objects demonstrate these geometric principles.

Encourage students to become “geometry detectives” who hunt for closed figures, regular polygons, and symmetrical objects in their environment. They might photograph examples, create a classroom display, or maintain a geometry journal documenting their discoveries. This real-world connection transforms abstract mathematical concepts into observable, meaningful patterns.

When students understand what makes figures special-whether they’re closed, regular, or symmetrical-they’re not just learning geometry. They’re developing spatial reasoning, pattern recognition, and analytical thinking skills that extend far beyond mathematics into art, science, architecture, and design.

What do you think? Can you find three objects in your classroom that are symmetrical? How would you explain to a friend what makes a square more “regular” than a rectangle?

References

1. https://www.splashlearn.com/math-vocabulary/geometry/open-shape
2. https://thirdspacelearning.com/us/blog/regular-irregular-shapes-explained/
3. https://www.hmhco.com/blog/folding-shapes-activities
4. https://www.innatepress.com/blog/teaching-symmetry-to-elementary-students