Properties of Shapes and Lines of Symmetry

Run your finger around a triangle. You follow three straight sides and return to your starting point. That makes it a polygon: a flat, closed shape with only straight sides.

Is the outline a polygon?

Check two things. Do all the sides join up, with no gap? Is every side straight? A square passes both checks. A circle fails the straight-side check. A semicircle has a straight side, but it also has a curved edge. It is not a polygon. An almost-finished triangle with a gap is open, so it fails the closed-shape check.

A vertex is a corner where two sides meet. More than one are called vertices. Each time you reach a new side, you pass a corner. A polygon has the same number of sides and vertices.

Give the shape its name

Squares and rectangles both belong to the quadrilateral group. The name tells you they have four sides. "Tri" can remind you of a tricycle's three wheels. "Oct" can remind you of an octopus's eight arms.

Count once around

Imagine a flat shape with seven straight sides of different lengths.

1. Mark one side with a small dot.
2. Follow the outline in one direction, counting each side.
3. Stop before you count the marked side again.
4. Count the corners to check. You should find seven.

Seven sides make a heptagon. The sides do not need to match in length.

Watch for corners that point inward. An L-shaped outline has six sides, so it is a hexagon. A four-sided arrowhead is a quadrilateral. A five-pointed star outline has ten sides: five corners point out and five point in. It is a decagon, not a pentagon.

Turning or flipping a polygon keeps its sides and corners. A square resting on one corner is still a square. You may hear someone call it a "diamond." Count and check its parts to find its math name.

Two shapes can have four sides and still be different. Look at the lengths of their sides and the sizes of their corners.

Compare the sides

Put a ruler's 0 mark at one end of a side. Read the length at the other end. Measure each side in the same unit, such as inches or centimeters. The ruler's end may not be its 0 mark, so check where you start.

For a rectangle with sides 6 cm, 3 cm, 6 cm, and 3 cm, opposite sides match. The two long sides match each other. The two short sides match each other. Small matching tick marks in a drawing also show equal sides.

Test a square corner

A square corner fits the corner of a sheet of paper. Its math name is a right angle. Place the paper corner inside the shape's corner. Both paper edges must line up with the shape's sides. A narrower or wider corner does not fit.

Corners can match without being square. To compare two corners, trace one on thin paper. Move the tracing onto the other corner. Their sides line up when the corners are the same size.

Use both checks for regular shapes

A regular polygon has all its sides equal and all its corners equal. A square passes both checks. So does an equilateral triangle, which has three equal sides. A regular stop-sign octagon has eight equal sides and eight equal corners.

Try a rectangle that is longer than it is wide. Its corners all match, but its long and short sides differ. It is irregular, meaning it is not regular.

Now try a rhombus with no square corners. All four sides match. Some corners are wider than others, so it is irregular too. Equal sides alone are not enough.

An irregular shape can still have matching folds. A long rectangle is irregular and has two lines of symmetry. A regular polygon has as many lines of symmetry as sides: a regular hexagon has six.

A property is something about a shape that you can check. Side counts, equal lengths, and square corners are properties. They help you describe a shape to someone who cannot see it.

Draw from a plan

You can draw a rectangle on squared paper. Start at a grid corner.

1. Draw four squares across with a ruler.
2. Draw two squares up.
3. Draw four squares back.
4. Draw two squares down to your starting point.

The outline is closed. The opposite sides match, and the four corners are square corners.

For a house-shaped pentagon, draw a square with a triangle roof. Make the roof's two slanted sides equal. Trace only the outside outline. The line under the roof is inside the picture, so it is not an outside side. You will count five sides and five vertices.

Give useful clues

Try this description: "My shape has four equal sides and four square corners." Those clues describe a square. "My shape has four sides" leaves many shapes to choose from. Adding a clue helps your partner decide.

A shape's color, size, or position does not change its kind. A tiny green square and a large tilted square are both squares.

Check both sorting rules

A Venn diagram has overlapping circles. Label one circle "has 4 sides" and the other "has a square corner."

• A square fits both rules. Put it in the overlap.
• A rhombus without square corners fits only the four-side rule.
• A triangle with a square corner fits only the corner rule.
• An equilateral triangle fits neither rule. Put it outside both circles.

A sorting table can use the same two questions. Give each question a "yes" and a "no" choice. Check both questions before choosing a box.

Is the rule true for the whole group?

"Every triangle has three vertices" works for every triangle, however it is turned. But "every quadrilateral has four equal sides" fails. A long rectangle is a quadrilateral, and its sides are not all equal. Finding just one shape that fails is enough to show the rule is false.

Fold a paper heart down the middle. The two halves land exactly on each other. The fold is a line of symmetry, also called a mirror line. A shape with a line like this is symmetric.

Try the fold test

Trace a shape on thin paper or cut out its outline. Choose a line to test. Fold along that line and press the paper flat. Check the sides and corners. Every part of one half must cover its partner on the other half. A corner sticking out shows that the line does not work.

A partner point is the same distance from the line, on the opposite side. A point on the line matches itself.

Try the mirror test

Stand a small mirror upright on your chosen line. Look at the half in front of it and the half you see in the mirror. Together, they must look exactly like the whole shape. A different-looking picture means that this line fails the test.

Equal pieces are not always mirror halves

Draw a line between opposite corners of a long rectangle. You get two triangles with the same size and shape. But fold along that line and parts stick out. The triangles do not land on each other. A line of symmetry needs a match by folding.

For pictures, check the inside as well as the outline. Draw a face with matching eyes and a red spot on only one cheek. The outline may match across the middle. The red spot has no partner, so the whole picture fails that symmetry test.

Look around you

A real butterfly, leaf, or face often has halves that match closely. Tiny differences make it nearly symmetric. A carefully drawn butterfly can match exactly.

Flags give you designs to test. Canada's flag has one vertical line through the maple leaf. Japan's centered red circle and white rectangle have two lines: one vertical and one horizontal. The United States flag has none. Its blue box of stars in the top left has no matching box on the other side of a possible fold.

A line of symmetry can point in any direction. Vertical means straight up and down. Horizontal means straight across. Diagonal means on a slant.

Test different directions

Lay a long paper rectangle flat, with its long sides across the page. Fold left onto right: the halves match. That gives a vertical mirror line. Unfold, then fold top onto bottom. That gives a horizontal mirror line. A corner-to-corner fold fails because the sides do not line up.

A square has those two matching folds too. It also matches when you fold between either pair of opposite corners. Those are its two diagonal lines of symmetry.

Turn the paper

Tilt the long rectangle. Its two mirror lines tilt with it. There are still two: turning changes where they point, not how many there are.

To test a slanted line, turn your paper until the line stands straight up and down. Now compare left and right, or fold along the line. Turn the paper back after checking. On squared paper, a diagonal line often goes through grid corners and cuts squares in half.

Check letters and words

Draw a block capital "T" with equal arms on its top bar. A vertical fold matches its halves. A horizontal fold does not: the top bar has no partner at the bottom.

Draw a block capital "B" with top and bottom bumps the same size. It matches across a horizontal middle line. Looking only for an up-and-down line would miss its symmetry.

Letters depend on how you draw them. Test your actual letter, including the lengths and shapes of its strokes. A "B" with a larger bottom bump will not pass the same fold test.

Words need matching letters and matching places. In "MOM," draw both "M" letters alike, with symmetric strokes, around a centered symmetric "O." A vertical fold pairs the two "M" letters and folds the "O" onto itself. For a horizontal fold in "BOX," each letter must match across the same horizontal middle line.

One matching fold may not be the only one. Test more directions before you finish counting. Draw each working line once with a ruler. A line has two ends, but it still counts as one line.

Compare triangles and four-sided shapes

For a triangle with three equal sides, each line joins a corner to the middle of the opposite side. For exactly two equal sides, the only working line starts where those equal sides meet.

A square has two lines through side middles and two through opposite corners. A long rectangle has only the side-middle lines. A non-square rhombus has its two diagonal lines.

A kite with two equal short sides and two equal long sides has one line. It joins the corners where the matching pairs meet. An isosceles trapezoid, with exactly one parallel pair and equal slanted sides, also has one. A slanted parallelogram with unequal neighboring sides has none.

Use the rule for regular polygons

A regular polygon has as many mirror lines as sides. A regular pentagon has five, a regular hexagon six, and a regular octagon eight.

For an odd number of sides, each line joins a corner to the middle of the opposite side. For an even number, half join opposite corners. The rest join opposite side middles. A regular hexagon has three of each kind.

Count sides carefully, then ask whether the polygon is regular. An L-shaped hexagon with equal arms has six sides but only one diagonal mirror line. The regular-polygon rule does not fit that L.

What about circles?

Every straight line through a circle's center is a line of symmetry. You can always draw another one. A circle has infinitely many lines of symmetry. A semicircle has just one, through its curved top and the middle of its straight side.

You can sort by symmetry too. Label circles "has 4 sides" and "has a line of symmetry." A square goes in the overlap. An equilateral triangle goes in the symmetry circle only. A triangle with no equal sides goes outside both.

A reflection is a same-size copy flipped across a mirror line. When you draw it beside the given half, the whole shape is symmetric in that line.

Count from a vertical line

Draw a vertical mirror line along a grid line. On its left, mark a corner two squares away. To find its partner:

1. Stay in the same row.
2. Count two square spaces to the mirror line.
3. Continue two square spaces on the right.
4. Mark the partner corner there.

Count spaces between grid lines, not the lines themselves. Start measuring at the mirror line, not at the edge of the page.

Complete a half shape

Try half a house. Mark A on the mirror line, four squares above the bottom. Mark B three squares left of the line and two squares above the bottom. Mark C three squares left at the bottom. Mark D on the mirror line at the bottom. Join A to B, B to C, and C to D.

A and D stay where they are because they lie on the line. Put B's partner three squares right, at the same height as B. Put C's partner three squares right, at the bottom. Join A to B's partner, then to C's partner, then to D.

The finished house has five corners. D lies in the middle of its straight bottom side, so it is no longer a corner of the whole outline.

Reflect across a horizontal line

Now count up or down. A corner three squares above a horizontal line needs a partner three squares below it. Keep the same column. Mark every partner before joining them in the same order as the original corners.

Flip shapes and match colors

A branch pointing away from the line to the left points right in the reflection. Sliding the drawing sideways would leave the branch pointing left. Check parts that touch the line: they must still touch it after reflecting.

Colored squares need same-color partners. A blue square touching the line on the left needs a blue square touching it on the right. Match the spaces from the line as well as the color. Fold the paper or use a mirror to check the finished pattern.

Your path crosses a slanted mirror line at a right angle, like the corner of a sheet of paper. Travel equally far on each side. On a square grid, your path may be diagonal too.

Follow the other diagonal

Draw a mirror line slanting up to the right through grid corners. Mark a dot one diagonal step down and right from a corner on that line.

Move the dot's path up and left, across one square. You reach the mirror line. Continue one more equal diagonal step up and left. That is the partner point. The path crosses the mirror line at a right angle.

Now mark a dot one square straight right of a corner on the line. Its partner is one square straight above that same corner. The up-left path between them meets the mirror line in the middle of a square. You travel half a diagonal step to the line and half a step beyond it.

Do not count straight across a slanted line as though it stood upright. Turn the paper until the line is vertical to check the flip. You can also trace the drawing and line on thin paper, then flip the tracing over along the line.

Fill four quarters

Draw one vertical and one horizontal mirror line that cross along grid lines. They split the grid into four quarters.

1. Make a pattern in the top left quarter.
2. Reflect it into the top right quarter.
3. Reflect the whole top half into the bottom half.

If your first quarter has three shaded squares, each needs a matching square in the other three quarters. The finished design has twelve shaded squares. Check both lines: every square needs the right color and place across each one.

Make your own design

Start by drawing a mirror line, then draw and color on one side. Reflect each part to finish your picture.

You can also fold paper, draw half a shape against the fold, and cut it out. Ask an adult for help with scissors. When you unfold it, the cut halves match. Fold twice, once each way, to make a design with two matching folds. Pressing paint inside a folded sheet makes a mirror print too.

A 3D shape, or solid, takes up space. It has length, width, and height. A face is a flat surface. An edge is a line where two faces meet. A vertex is a corner point where edges meet.

Count a cube in groups

Hold a cube and find its square faces. There is one on top, one underneath, and four around the sides: six faces.

Next, trace its edges. Count four around the top, four around the bottom, and four joining top to bottom. That makes twelve edges.

Count four vertices at the top and four at the bottom: eight vertices. A small sticker can mark each part after you count it. An edge is a line; a vertex is a point.

Compare familiar solids

All six cube faces are equal squares. A rectangular prism has rectangular faces; some can be squares. A cube is a special rectangular prism.

A right prism has two matching polygon ends joined by rectangles. For a triangular prism, count two triangle ends and three rectangle faces. Count three edges around each end, then three joining edges: nine. There are three vertices at each end: six. A right prism with pentagon ends has five rectangle faces plus two ends, making seven faces.

A pyramid has a polygon base and triangles meeting at one tip. A square pyramid has one square base and four triangle faces. Count four base edges and four rising to the tip: eight. Count four base corners and the tip: five vertices. A triangular pyramid has four triangle faces.

Include the hidden parts

Turn a solid as you count. In a drawing, some parts are hidden behind it. Dashed lines may show hidden edges. A cube still has six faces even when only three appear in the picture.

Two faces share each edge of these solids. Counting all four sides of every cube face gives twenty-four, because you count every edge twice. Trace the edges on the whole solid to count each one once.

A ball and a can have curved parts that a box does not have. Feel or imagine the surface bending around each one.

Compare curved solids

The cone's tip is its vertex, also called its apex. Its circular edge is where the flat face meets the curved surface. A sphere has no joining edge and no corner.

A cylinder rolls on its curved surface. Stand it on a flat circular face and you can stack another cylinder on top. A cube or rectangular prism also has a flat top and bottom for stacking. A cone or pyramid standing on its base has a pointed top.

Find the flat shapes

Press a cube face into modeling clay or trace around it. Its mark is a square. A cylinder's flat face makes a circle. A square pyramid gives one square and four triangles. A right triangular prism gives two triangles and three rectangles.

Turning a solid keeps its name and parts. A cylinder lying down is still a cylinder. An upside-down pyramid is still a pyramid. A number cube is close to a cube, a cereal box to a rectangular prism, and a ball to a sphere.

Build an edge frame

Use one stick for each edge and one small clay ball for each vertex. A cube needs twelve sticks and eight balls. A square pyramid needs eight sticks and five balls. A triangular prism needs nine sticks and six balls. The sticks show where faces would meet; they do not fill in the faces.

Sort using two rules

Make overlapping circles labeled "has a curved surface" and "has a vertex."

• Put a cone in the overlap: it has both.
• Put a cylinder and sphere in the curved-surface circle only.
• Put cubes, prisms, and pyramids in the vertex circle only.

Compare solids one property at a time. A cube and square pyramid both have a square face. Their other faces and their edge and vertex counts differ.