2D and 3D Shapes: Sides, Vertices, Faces and Edges

You can draw a circle and hold a ball. The circle and the ball are related, but they have different shape names.

Shapes you can draw

A flat shape, or 2D shape, has length and width. It has no thickness. A circle, triangle, square, and rectangle are flat shapes. You can follow their outlines on a page with your finger.

A solid shape, or 3D shape, takes up space. It has length, width, and height. You can turn a solid over and look at another part. A ball, a box, and a can are solid objects.

A paper cutout has a tiny thickness, so the paper itself is solid. When you name its triangle shape, you are looking at its flat outline.

Give the whole shape its name

A ball is shaped like a sphere. A game die is shaped like a cube. A cereal box is shaped like a rectangular prism.

The flat shape on an object can have a different name. Each flat surface of a cube is a square. The top of a can is a circle. Say “square” for that flat shape and “cube” for the whole solid.

Sort a small collection

Imagine a circle drawing, a square drawing, a soccer ball, a cereal box, and a soup can.

1. Look at the circle and square drawings. Each shows a flat shape.
2. Look at the ball, box, and can. Each is a solid.
3. Count the groups. There are 2 flat shapes and 3 solids.

Do not call the whole ball a circle. A circle names a flat round shape; a sphere names the whole round solid.

Read a picture of a solid

A drawing can show a solid even though the page is flat. Lines slant back to show depth. Dashed lines may show edges hidden behind the solid.

When a picture shows a box, name the shape it shows. It shows a rectangular prism. You can imagine turning that box to see its back and bottom.

A triangle can be tiny, huge, green, or blue. To decide whether it is a triangle, check its sides.

Check the rules

A triangle is closed and has exactly 3 straight sides. Closed means the sides join all the way around. There are no gaps. Straight means a side does not bend.

These rules are called defining attributes. They are the things a shape must have to get its name. You can ask, “Do the sides join?” and “Are all 3 sides straight?”

An open drawing with 3 straight lines is not a triangle. A drawing with 2 straight sides and a curved part is not a triangle either. Each drawing breaks one triangle rule.

Turn it and check again

Turn a triangle until it points down. Its sides still join up. It still has 3 straight sides, so it is still a triangle.

A triangle does not need equal sides or a flat bottom. A long, thin triangle follows the same rules as a wide one.

A square keeps its name when it stands on one corner. You may hear someone call that shape a “diamond.” Its math name is still square. Tilting the page can help you see its sides and corners clearly.

Things that can change

Color, size, material, and position do not change a shape's name. Thick pencil lines do not change the name either. A red wooden square and a blue paper square follow the same square rules.

Now check these drawings:

• A red triangle pointing down: its sides join, and all 3 are straight. It is a triangle.
• Three straight lines with a gap: the outline is open. It is not a triangle.
• Two straight sides joined by a curve: the outline closes, but one part bends. It is not a triangle.
• A long, thin triangle: its sides join, and all 3 are straight. It is a triangle.

Draw by following rules

Draw 3 straight sides that meet to make a closed shape. Start at one corner and trace the outline. Check that you return to the start without crossing a gap.

Your triangle may look different from a friend's triangle. Both drawings are correct when both follow the triangle rules.

A shape's outline tells you where its sides and corners are. Follow it slowly so you count each part once.

Sides and vertices

A side is a line along the outline of a flat shape. A vertex is a corner where sides meet. Two or more corners are called vertices. You can say “corners” while you learn the new word.

Put a small mark at one corner. Move around the outline and count each side. Stop when you reach your mark again. Then trace around a second time and count the vertices.

A long side counts as one side. A short side counts as one side too. A line inside a rectangle does not add a side to its outside outline. It marks smaller shapes inside the rectangle.

Count corners that point in

A pentagon has 5 sides and 5 vertices. For a closed shape made only of straight sides, these two counts match. Each side ends where the next side begins.

A star outline needs careful counting. A 5-pointed star has 5 outer tips and 5 inward corners. It has 10 vertices altogether, and 10 sides. Trace into each dip as well as out to each tip.

Curved outlines

A circle has no straight sides and no vertices. Its outline is one curved line. The point in its middle is its center. Every point on the outline is the same distance from that center.

You can draw a circle with a pencil tied to a string. Keep the other end fixed at the center. Keep the string tight as the pencil goes around. The string keeps the distance the same.

An oval looks stretched or egg-like. It has no straight sides and no vertices. Its outline is not the same distance from the middle all the way around.

A semicircle is half a circle. Follow its outline: one part is straight and one part curves. It has 1 straight side and 1 curved side.

Which shapes are polygons?

A polygon is a closed flat shape made only of straight sides. A triangle and a pentagon are polygons. A circle, oval, and semicircle are not polygons because their outlines curve.

Check both rules. Straight lines with a gap do not make a polygon. A closed outline with a curved part does not make one either.

You can name many flat shapes by counting sides. For four-sided shapes, check the corners and side lengths too.

Start with three or four

A triangle has 3 straight sides and 3 vertices. Its sides can be different lengths. Turning it does not change those counts.

A quadrilateral is a closed shape with 4 straight sides. It has 4 vertices too. The word begins with “quad,” which means four. Squares, rectangles, rhombuses, and trapezoids belong to this group. A lopsided four-sided shape belongs too.

Check for square corners

A square corner looks like the corner of a sheet of paper. Fit a paper corner into the corner you want to check. It fits exactly when there is no gap or overlap.

A rectangle has 4 straight sides and 4 square corners. Opposite sides are the sides across from each other. In a rectangle, opposite sides have the same length.

A square has 4 equal sides and 4 square corners. It follows every rectangle rule. That makes every square a special kind of rectangle. A long rectangle has 4 square corners, but its sides are not all equal.

Look at the sides across from each other

Parallel sides stay the same distance apart. Imagine making them longer and longer: they would never meet. Train tracks help you picture this.

A rectangle has 2 pairs of parallel sides. One pair is across the top and bottom. The other pair is across the left and right. The pairs stay parallel when you turn the rectangle.

A rhombus has 4 equal sides. Its corners do not have to be square. Some look like pushed-over squares. A square fits the rhombus rule too because all 4 sides are equal.

A trapezoid has a pair of parallel sides. The red trapezoid in many pattern-block sets shows this pair clearly.

Check a group of four shapes

Look at a square, a long rectangle, a rhombus without square corners, and a lopsided quadrilateral.

All four are quadrilaterals: each has 4 straight sides. Only the square and long rectangle have 4 square corners. Those two are rectangles.

A square tipped onto a corner still passes both square checks. Its equal sides and square corners stay the same.

A polygon's sides do not need to be equal. Count them to find its name, even when the outline looks unusual.

Use the side count

Each of these polygons has the same number of vertices as sides. A nonagon has 9 sides and 9 vertices.

Two word clues can help. “Hexagon” and “six” both have an x. An octopus has 8 arms, and an octagon has 8 sides.

A shape can have a dent

A hexagon can have unequal sides. It can also have a corner pointing inward. Trace the outline of a thick capital L. Count all the way around, including the corner inside its bend. You count 6 straight sides and 6 vertices. That outline is a hexagon.

Do not count just the sides you notice first. A small side beside a long one still counts. Mark a start and follow the whole outline.

When is a polygon regular?

A regular polygon has all sides the same length and all corners the same size. A square is a regular quadrilateral. A long rectangle has square corners, but unequal side lengths, so it is not regular.

A stop sign has a regular octagon outline. A honeycomb cell is shaped like a regular hexagon. Home plate on a baseball field has a pentagon outline. These objects can help you picture the shapes, and counting checks the names.

Use a corner count

Suppose a polygon has 7 vertices. Its side count matches its vertex count, so it has 7 sides. Look up 7 in the table: it is a heptagon.

A hexagon has more vertices than a pentagon. A hexagon has 6, while a pentagon has 5.

Draw your own pentagon

Draw 5 straight sides that join into one closed outline. A house outline with a pointed roof is one way. Count around its outside: 5 sides make a pentagon. Your pentagon can be wide, narrow, or uneven and still have the right name.

Hold a cube-shaped block. You can lay a finger on a flat part, trace a line, or touch a corner. These parts have different names.

Touch each kind of part

A face is a flat surface of a solid. A curved surface bends, like the outside of a ball.

An edge is a line where two surfaces meet. Trace an edge with your finger. A vertex is a corner or pointed tip. Touch one with a fingertip. The plural is vertices.

Edges meet at the vertices of a cube or pyramid. A cone has a pointed tip too. We count that tip as 1 vertex, even though no edges meet there.

Use “sides” for a flat shape's outline. A square has 4 sides. A cube has 6 faces and 12 edges.

Count a cube in groups

Imagine the cube sitting on a table.

1. Count faces: 1 on top, 1 underneath, and 4 around the sides. That makes 6 faces.
2. Count edges: 4 around the top, 4 around the bottom, and 4 joining them. That makes 12 edges.
3. Count vertices: 4 on top and 4 on the bottom. That makes 8 vertices.

Each kind of part gets its own count. Counting 8 corners tells you the number of vertices, not edges.

Find the hidden parts

A cube drawing often shows just 3 faces. The cube still has 6. Imagine the back and bottom, or turn a real model to see them.

Small removable stickers help with face counts. Mark a face, count it, and turn the solid. Stop when every face has a mark. For edges, count in groups around the ends and between them.

Find flat shapes on solids

Each face has a flat shape. A cube has square faces. A pyramid has triangle side faces. A cylinder has circle faces.

Press a flat face gently into play dough, or trace around it. The mark shows the face's shape. The solid and the flat mark have different names.

Two ways books use “face”

Here, faces are flat. Count a curved surface separately. A cylinder has 2 flat faces and 1 curved surface.

Some books say “curved face.” They count the cylinder's 2 flat faces and 1 curved face as 3 faces altogether. The cylinder has not changed. The words tell you which parts to include.

A box can stand tall or lie on its side. Its faces, edges, and vertices stay the same. Use those parts to recognize its solid shape.

Two matching ends

A prism has two matching flat ends facing each other. Flat faces join the ends. The prisms here have rectangle faces around the sides.

A prism gets its name from its ends. A triangular prism has triangle ends. A hexagonal prism has hexagon ends. Look for both ends even when one is hidden.

Cubes and rectangular prisms

A cube has 6 square faces, all the same size. It has 12 edges and 8 vertices. A game die and a sugar cube have this shape.

A rectangular prism has 6 rectangle faces. Opposite faces match. It also has 12 edges and 8 vertices. A cereal box, brick, and closed book are shaped like rectangular prisms.

A square is a special rectangle. In the same way, a cube is a special rectangular prism. All its faces are squares. Face, edge, and vertex counts alone cannot tell a cube from a longer rectangular prism. Check the face shapes too.

Count a triangular prism

Find one triangle end, then the matching end. There are 2 triangle faces. Three rectangle faces join them, making 5 faces altogether.

Each triangle end has 3 vertices. Count 3 at one end and 3 at the other: 6 vertices.

Now count edges in three groups:

• 3 around one triangle end;
• 3 around the other triangle end;
• 3 joining matching corners of the ends.

That makes 9 edges. A tent can help you picture a triangular prism resting on a rectangle face.

Count a hexagonal prism

An unsharpened six-sided pencil helps you picture this prism. Its two ends are hexagons. Six rectangle faces run between them.

Count the faces: 2 ends and 6 side faces make 8. Each end has 6 vertices, making 12 vertices. There are 6 edges around each end and 6 joining them, making 18 edges.

Look for a solid whose triangle faces meet at one point. Then find the face opposite that point. You have found a pyramid's base.

One base and an apex

A pyramid has a flat base and triangle side faces. The side faces meet at one point called the apex.

Name a pyramid after its base. A square-based pyramid has a square base. A triangular-based pyramid has a triangle base. The base is part of the solid, even when it rests on a table.

Count a square-based pyramid

Start with the faces. There is 1 square underneath and 4 triangles around it. That makes 5 faces. Counting just the triangles misses the base.

Next, count the edges. There are 4 around the square base. Another 4 join the base's corners to the apex. That makes 8 edges.

For vertices, touch the 4 base corners, then the apex. That makes 5 vertices. The apex counts once, even though several triangle faces meet there.

A pyramid with triangle faces all around

A triangular-based pyramid has 4 faces, and all are triangles. Its other name is tetrahedron.

Its triangle base has 3 vertices. Add the apex to get 4 vertices. Count 3 edges around the base and 3 up to the apex: 6 edges.

Any one of its triangle faces can be its base. Turn the solid to rest on another face. It is still a triangular-based pyramid.

Compare a pyramid and a prism

A cube and a square-based pyramid both have a square face. The cube has 6 faces, 12 edges, and 8 vertices. The pyramid has 5 faces, 8 edges, and 5 vertices. Its triangle side faces rise to the apex.

A flat triangle is not a pyramid. A triangle is one flat shape; a pyramid is a whole solid. A pyramid keeps its name when you turn it upside down. Its base still names it, even when that base is no longer underneath.

A ball, a can, and a party hat all have a curved surface. Look for flat faces and pointed tips to tell their shapes apart.

A sphere is round all around

A sphere is a perfectly round solid. Every point on its surface is the same distance from its center. A ball, marble, or globe is shaped like a sphere.

A sphere has 1 curved surface. It has no flat faces, edges, or vertices. A circle is flat; a sphere is the whole round solid.

A cylinder has two circle faces

A cylinder has 2 matching flat circle faces and 1 curved surface. A soup can or paper towel roll has this shape.

Trace around each circle face. Where the flat face meets the curved surface, there is a curved edge. A cylinder has 2 curved edges and no vertices. An edge can curve; it does not have to be straight.

A cone has a pointed tip

A cone has 1 flat circle face and 1 curved surface. The curved surface narrows to the apex. We count that pointed tip as 1 vertex.

A cone has 1 curved edge around its circle face. A party hat or an ice cream cone helps you picture this solid.

A cone is different from a pyramid. The cone has a circle base and a curved surface. The pyramid has a base with straight sides and flat triangle faces.

Half of a sphere

A hemisphere is half a sphere. Imagine a solid ball cut in half through its center. Each half has 1 flat circle face, 1 curved surface, and 1 curved edge. It has no pointed tip or corners.

Compare a cylinder and a cone

Both have 1 curved surface. The cylinder has 2 circle faces and no tip. The cone has 1 circle face and 1 tip. You can tell them apart without using their size or color.

When a book counts curved surfaces as faces, it gives different face totals: 3 for a cylinder, 2 for a cone, and 1 for a sphere. Look for the words “flat face” or “curved face” so you know what is counted.

Look around for something round, something box-shaped, and something with a pointed top. You can describe each object with a solid shape name.

Match an object to a solid

A ball is shaped like a sphere. A game die is cube-shaped. A cereal box is shaped like a rectangular prism. A soup can is shaped like a cylinder. Many tents are shaped like triangular prisms, and a party hat is shaped like a cone.

Real things are not always perfect math shapes. An orange is roughly a sphere. A die may have rounded corners. Choose the shape that fits the whole object best.

You can find flat shapes on objects too. A window can show a rectangle. A clock face can show a circle. A stop sign has an octagon outline. Each face of a cube-shaped die is a square.

A pizza slice with a curved crust is not a triangle. The curved part breaks the rule that all 3 sides must be straight.

Roll, slide, or stack

A curved surface lets a solid roll. A sphere rolls in every direction. A cylinder on its curved surface rolls along a straight path. A cone on its curved surface rolls around in a circle.

A flat face lets a solid slide along a surface. Cubes and rectangular prisms slide on their flat faces. A cylinder can slide when it stands on a flat face.

To stack easily, an object needs a flat top and bottom. Cubes, rectangular prisms, and upright cylinders have these. A cone or pyramid has a pointed top, so another block cannot sit flat on top of that point.

Change how a can rests

Stand a can on its flat circle face. Its top is flat too, so you can stack another matching can on it.

Now lay the can on its curved surface. Give it a gentle push. It rolls. The shape stays a cylinder; the part touching the table has changed.

Match the shape to the job

Boxes shaped like rectangular prisms pack closely together and stack on shelves. Round wheels roll smoothly. Sphere-shaped balls roll the same way in every direction.

On a table, place a cereal box, soup can, orange, and game die. The can and orange can roll. The box, die, and upright can can stack. Check the part touching the table before you predict what each object will do.

You can sort the same shapes in several ways. Say your sorting rule first so every shape has a clear place.

Ask a yes-or-no question

A sorting rule is a question you can answer “yes” or “no.” Try “Does it have 4 straight sides?” Put each shape in the yes group or the no group.

Check one shape at a time. A square has 4 straight sides, so it goes in yes. A triangle has 3, so it goes in no. A circle has no straight sides, so it goes in no too.

After sorting, count both groups. Together they should include every shape you started with. A shape must not be left out or counted in both groups.

Work through a collection

Sort a triangle, square, circle, hexagon, oval, and rectangle. The rule is “Has only straight sides.”

1. Check the triangle, square, hexagon, and rectangle. All their sides are straight. Put these 4 shapes in yes.
2. Check the circle and oval. Their outlines curve. Put these 2 shapes in no.
3. Check the total: 4 shapes and 2 shapes make 6 shapes.

Color can be a sorting rule too. But sorting red from blue is sorting by color. It does not tell you about the sides or surfaces that give a shape its name.

Change the rule

Try a cube, sphere, and cylinder. With “Is it a solid?” all three go together. With “Does it have a curved surface?” the sphere and cylinder go in yes. The cube goes in no.

The shapes have not changed. Your question has changed which property you check.

Find a hidden rule

One group holds a cube, rectangular prism, and triangular prism. Another holds a sphere, cylinder, and cone.

Look for something shared by every shape in one group. The second group all have a curved surface. None in the first group do. “Has a curved surface” fits the whole collection.

An odd-one-out puzzle may allow different reasons. From a square, rectangle, triangle, and circle, choose the circle for its curved outline. Or choose the triangle because only it has exactly 3 straight sides. Name the property that supports your choice.

One shape can follow two rules at the same time. You need a place that shows both answers.

Overlapping circles

A Venn diagram uses two circles that overlap. Give each circle a sorting rule. Check both rules for each shape.

• First rule only: put it in the first circle, away from the overlap.
• Second rule only: put it in the second circle, away from the overlap.
• Both rules: put it in the overlap.
• Neither rule: put it outside both circles.

A shape goes in one place. The overlap already shows that it follows both rules.

Place solids in a Venn diagram

Let the first rule be “Has a flat face.” Let the second be “Has a curved surface.”

A cube, triangular prism, and square-based pyramid have flat faces but no curved surfaces. They go in the first circle only. A sphere has a curved surface but no flat faces. It goes in the second circle only.

A cylinder and a cone have both kinds of surface. They go in the overlap. Do not put either solid in just one circle: that would leave out one of its properties.

Use a sorting grid

A sorting grid, also called a Carroll diagram, uses rows and columns. One rule labels the rows. The other labels the columns. Each shape fits in exactly one box.

Try these rows: “4 sides” and “Not 4 sides.” Try these columns: “All sides equal” and “Not all sides equal.”

The regular octagon does not have 4 sides, but all 8 sides match. It belongs in the bottom-left box. A triangle with three different side lengths fits the bottom-right box.

Compare using more than one clue

A square is flat, with 4 sides and 4 vertices. A cube is solid, with 6 faces, 12 edges, and 8 vertices. Each cube face is a square.

Two solids can share one count. A square-based pyramid and a triangular prism both have 5 faces. The pyramid has 8 edges and 5 vertices. The prism has 9 edges and 6 vertices. Check the faces' shapes or another count to tell these solids apart.