Shape Definition, Properties & Worked Examples

Everything in the world has a shape definition. Any two-dimensional or three-dimensional object has an outline which is its shape. Students learn about these geometric shapes throughout the elementary grades to help them recognize shapes and their attributes in real life.

This blog post covers the English definition of shape, the different types of shapes, examples of shape, and sample shape problems for elementary students.

What is shape definition?

Shape is defined as the outline of an object. There are many different shapes in the world, big and small.

Some shapes are more familiar than others, but everything around us has a shape. For example, the outline of a hand has a curved shape with no defined edges or vertices. While a table top has a rectangular shape with measurable angles and side lengths.

Few shapes are one-dimensional, such as a point or line segment. Other geometric shapes are two-dimensional such as a triangle or square. However, many shapes in the world around us are three-dimensional shapes. These are composed of various 2D-shaped faces to make a tangible 3D shape.

Regardless of their dimension, all shapes have attributes that help categorize them. For example, curved sides or straight sides, square or rectangular faces.

Different types of shape

Giving names to math shapes based on characteristics allows for shape categorization. Shapes can be categorized into three main groups.:

• Open shapes and closed shapes
• Geometric shapes and non-geometric shapes (organic shapes)
• Two-dimensional shapes and three-dimensional shapes

Many examples of shape overlap between these groups. For example, an octagon is a closed shape which is also a two-dimensional shape and a geometric shape.

Open shapes

One of the most basic ways to categorize shapes is whether they are open shapes or closed shapes.

Open shapes are shapes where the line segments or curved lines do not connect at their ends. Imagine the shape of a rainbow, the long curved line is an open shape.

To help students understand open shapes, teachers could explain that it’s like walking around a square desk but stopping before returning to the starting point. Open shapes have different starting points and ending points.

Here are more examples of open shapes:

Closed shapes

Closed shapes are shapes that start and end at the same point and are composed of line segments or curved lines that connect. There is no open space in the outline of a closed shape. The outline of an object can be curved or straight but as long as the outline connects, it is a closed shape.

This type of shape can be regular or irregular and are categorized as polygons and non-polygons based on curved or straight lines.

Some examples of closed shapes include:

• Circle
• Pentagon
• Quadrilateral
• Trapezoid
• Hexagon
• Oval

More examples of closed shapes

Teachers can support students in building an understanding of open and closed shapes with a string or piece of yarn. Teachers can have students make a shape using the string and connect one end to the other, for example in the shape of a circle. This is an example of a closed shape. Then they can have the student make a design using the string where the two ends do not connect. This is an open shape. This hands-on activity is a great way to build a conceptual understanding of open and closed shapes.

Geometric shapes

Geometric shapes are closed shapes that have measurable attributes. They can have curved lines or straight lines. Students learn to identify both curved line geometric shapes and straight line geometric shapes in the early grades.

For example, they identify circles, ovals, squares and pentagons. However, they learn to measure straight-line closed shapes (otherwise known as polygons) before they learn to measure curved-line geometric shapes.

Third and fourth graders learn to measure the perimeter and area of rectangles. Whereas, sixth and seventh graders learn to measure the circumference of circles.

Here are some examples of typical geometric shapes.

Circle

A circle is a closed, geometric shape. It is round with no corners or edges. Its measurable attributes are:

• Radius
• Diameter
• Circumference

Circles are an example of a closed shape that is not a polygon. Other closed curved shapes include ovals and ellipses.

Square

A square is also a closed, geometric shape. It has straight line segments that form the sides as well as the corners or vertices.

Unlike circles, squares are a type of polygon. It is a specific type of polygon called a quadrilateral, meaning it has four sides and four vertices. This particular shape is a special kind of quadrilateral because it has four congruent sides; all four sides are the same length. Consequently, all four angles have the same measure, 90 degrees.

Sometimes students think that rectangles and squares are synonyms. Despite the fact that both shapes are quadrilaterals, this is not true, a square is always a rectangle, but a rectangle is not always a square.

Other examples of shapes considered quadrilaterals include:

• Rectangles
• Parallelograms
• Trapezoids
• Rhombuses
• Kites

Dimensional shapes

Shape vs. form

Shape and form are often used as synonyms. However, shape and form describe different attributes.

The shape of an object describes its two-dimensional properties, length and width, or height and width. This describes the area that the outline of an object or shape covers in two dimensions.

The form of an object is not limited to two dimensions, it describes the three-dimensional volume the shape covers: length, width and height or depth.

For example, a house has form. It is a three-dimensional figure composed of various shapes to create a three-dimensional shape.

A cube also has form because it can be measured in three dimensions. However, a square has shape, not form because it is flat and can only be measured in two dimensions.

However, a square has shape, not form because it can only be measured in two dimensions and is flat. Teachers can help students understand shape vs. form by comparing a rectangle to a rectangular prism. It helps to have a real box or hands-on rectangular prism. Show students how to measure the rectangle with length and height, and how to measure the box with length, width and height.

Exploring this concept in a hands-on way helps students build conceptual understanding.

Two-dimensional shapes

Two-dimensional shapes lay flat and cannot be held because they do not have depth. Students learn to measure the length and height of a two-dimensional shape.

Students learn about many two-dimensional figures in the elementary grades, beginning with the most basic shapes. They learn about non-polygons such as circles, and polygons such as triangles.

As students progress, they learn that the name of each particular shape has meaning and the prefix of many shape names describes the number of sides the shape has. Many words in the English language are based on words from another language such as Greek or Latin. Learning the meaning of each prefix for geometric shapes helps students to understand and remember the number of sides each shape has.

Triangle

A triangle is a polygon with three sides. The prefix tri comes from Greek and Latin root words and means three.

Students learn about equilateral triangles, isosceles triangles and scalene triangles. They also learn to categorize types of triangles based on angle measures. For example, right triangle, acute triangle and obtuse triangle.

Students can remember that a triangle has three sides by remembering that a tricycle has three wheels because the prefix tri means three.

Quadrilateral

Quadrilaterals are polygons with four sides. The prefix quad comes from a Latin root word and means four. Quadrilaterals are the most common polygon that elementary students learn about.

Students learn to identify and categorize parallelograms, rhombuses, rectangles, squares and trapezoids based on angle measure and the presence or absence of parallel lines.

Pentagon

A pentagon is a polygon with five sides. The prefix pent or penta- comes from Greek words meaning five. Pentagons are often used to show students regular vs. irregular polygons. Regular polygons have all sides the same length (much like an equilateral triangle or a square). All interior angles have the same measure.

An irregular polygon has sides of different lengths and angles of different measures.

Hexagon

Hexagons are polygons with six sides. The prefix hexa- comes from the Greek word hex which means six. Hexagons are often represented in the shape of a honeycomb.

This trend of polygon names continues with seven-sided heptagons, eight-sided octagons, nine-sided nonagons, ten-sided decagons and beyond. These polygons are two-dimensional shapes because they do not have depth.

Three-dimensional shapes

Three-dimensional shapes have definite form. You can hold or touch 3D shapes because they have three dimensions: length, height and width or depth.

Some three-dimensional shapes are small like an ice cream cone. Other three-dimensional shapes are large, such as the sphere of the Earth. Some three-dimensional shapes have a rounded external surface such as spheres and cylinders while others have external surface faces that meet to form edges.

Learning the attributes that define each three-dimensional shape helps children to understand, identify, categorize and measure these three-dimensional geometric shapes. The most common three-dimensional shapes students need to describe are:

• Cubes
• Cones
• Cylinders
• Spheres
• Prisms
• Pyramids

Prisms and pyramids are two subcategories of three-dimensional shapes. These subcategories have variances.

Prisms

Elementary students learn about two different types of prisms, a rectangular prism and a triangular prism. However, there are also pentagonal, hexagonal and octagonal prisms (and others).

The type of prism describes the two-dimensional shape of the two bases. All other side faces of a prism are rectangles. When put together, these bases and faces create a three-dimensional shape.

For example in the hexagonal prism below, the bases are hexagons. The other faces are rectangles. Because a hexagon has six sides, this prism has two hexagonal bases and six rectangular faces.

Pyramids

Students learn about triangular, rectangular, square and hexagonal pyramids. However, the most commonly taught are triangular and square pyramids. The type of pyramid describes the shape of the base. All other side faces of a pyramid are triangles.

To help students build an understanding of two-dimensional vs. three-dimensional shapes, teachers can have students go on a scavenger hunt around the room. A teacher can post pictures or have objects of various shapes around the room and students go around with a checklist of each shape to keep track of the shapes they see. This is also a great way to connect shapes to real life and have students see shapes in their environment.

When do students learn about shape?

Students learn about shapes throughout elementary school. They start with identifying basic shapes and the properties of shapes. Firstly with two-dimensional geometric shapes in kindergarten and first grade. They then learn about three-dimensional geometric shapes, categorizing shapes based on specific attributes such as the number of sides and vertices in second and third grade. This helps students compare 2D shapes with 3D shapes.

As students learn more about different shapes they learn that there is a hierarchy of categories for certain shapes, based on characteristics. Once students learn to identify and categorize shapes, they can measure shapes and use them for other domains of mathematics, such as the area model for multiplication.

In fourth, fifth and sixth grade, students measure the perimeter, area and volume of various shapes including rectangles and later triangles.

The chart below demonstrates students’ progression of understanding shape in the geometry domain throughout elementary school. This is based on the common core state standards for mathematics.

Kindergarten & 1st Grade	2nd & 3rd Grade	4th Grade	5th & 6th Grade
– Name basic shapes – Identify 2d and 3d shapes – Compose larger shapes from basic shapes – Distinguish between defining and non-defining attributes – Create composite shapes – Partition shapes	– Recognize and draw shapes with specific attributes – Partition shapes – Measure the length of shapes – Categorize shapes and polygons – Measure the perimeter of quadrilaterals – Find the area of rectangles and squares	– Draw points, lines, rays and angles – Classify shapes based on angles and the presence or absence of parallel/perpendicular lines – Recognize lines of symmetry – Measure angles – Use area and perimeter formulas for rectangles	– Classify 2d shapes based on a hierarchy of properties – Measure the volume of rectangular prisms – Find the area and volume of rectangular prisms – Draw polygons on the coordinate plane given coordinates for vertices – Represent 3d shapes using nets

As students progress from identifying shapes to categorizing shapes to measuring shapes they need to pay attention to the shape’s characteristics. Teachers can support students with this concept by having them play ‘Guess my shape.’

In this game, a teacher describes a shape attribute by attribute. As they list each attribute, students draw the shape on a whiteboard or paper, adjusting it after hearing each clue. Then students try to name the shape as specifically as possible. This can also be played with a student as the leader.

For example, “My shape has four sides. My shape has two pairs of parallel sides. My shape does not have any 90° angles. All sides of my shape are the same length.” The answer in this example is a rhombus.

Examples of shape problems

Kindergarten

Question 1

Draw a closed shape. Does your closed shape have straight or curved lines?

Answer: Drawings will vary but should show evidence of a closed figure which demonstrates that students understand a closed shape has the same beginning and ending point.

First grade

Question 2

Which geometric figures are two-dimensional?

1. Octagon
2. Sphere
3. Cone
4. Rectangle
5. Quadrilateral
   

Answer: 1, 4, 5.

Octagons, rectangles and quadrilaterals are all two dimensionsal. Spheres and cones are three-dimensional. Teachers can encourage students to consider which shapes they could hold in their hand (sphere and cone) vs. which shapes they would just see on paper (octagon, rectangle, quadrilateral).

Second grade

Question 3

Of the shapes shown below, which shapes are polygons?

Answer: B and D. These particular shapes have straight sides and are closed shapes. Even though D is an irregular polygon, it is still a polygon (a pentagon) because it meets the criteria for a polygon. A, C and E are curved shapes so they are not polygons and F is an open shape so it is also not a polygon.

Third grade

Question 4

Which of the following shapes is a parallelogram but not a rectangle?

Answer: B and F are both parallelograms because they are quadrilaterals with opposite sides congruent and parallel but they are not rectangles because they do not have 90-degree angles. A and E have 90-degree angles so they are considered rectangles. C does not have two pairs of parallel sides. It is a trapezoid, not a parallelogram.

Fourth grade

Question 5

How many more sides does an octagon have than a hexagon?

Answer: 2 more sides because an octagon has 8 sides and a hexagon has 6 sides.

Fifth grade

Question 6

Look at the rectangular prism. What are the three dimensions of this shape that you could measure?

Answer: You can measure the length, height and width of this rectangular prism. Knowing these three dimensions can also help you measure the volume of the three-dimensional geometric shape by using the formula length x width x height.

Shape practice problems

When students learn about math shapes they need to solve problems involving shape identification and measurement that they might see in the real world.

The problems below include basic shapes that elementary students learn about and frame them in real-world situations.

Teachers can encourage students to draw visual models when problem solving with different shapes.

Kindergarten

Question 1

Mrs. Winters bought a can of beans at the store. What geometric shape name describes the shape of her can of beans?

• Cone
• Cylinder
• Prism
• Pyramid
• Sphere

Answer: B cylinder. A cylinder is one of the basic shapes that students learn about when learning three-dimensional shapes. A can of food is a great example of a cylinder in the real world.

Teachers can support students in understanding various three-dimensional geometric shapes by providing hands-on examples such as cans for cylinders and balls for spheres.

First grade

Question 2

Albert looked at the two shapes below and argued that shape B is not a triangle because it is upside down. His sister Ariella wanted to convince Albert that both shapes are triangles. What should Ariella say to Albert to convince him that both shapes are triangles?

Answer: Answers will vary. However, answers should focus on the defining attributes of a triangle vs. the non-defining attributes of a triangle. A triangle is a three-sided polygon. It has three sides and three vertices. The orientation, size and color of the triangle does not change the fact that it is a triangle. So even though shape B is in a different orientation than shape A, it is still a triangle.

Second grade

Question 3

The Summer Triangle is a pattern of three stars that form a triangle in the night sky. All three sides of the triangle are slightly different lengths. What is the name for this type of triangle?

Answer: A scalene triangle.

By definition, a scalene triangle is a three-sided polygon with all three sides different lengths and all three angles with different measure. The Summer Triangle is known as a ‘near isosceles triangle’ which means that two of its size are almost the same length, therefore it is a scalene triangle.

Third grade

Question 4

Dominique was packing to move into a new house. The size of her moving box was 3 feet long by 3 feet tall by 3 feet wide. What is the volume of her packing box? What geometric shape name could you use to describe her box?

Answer: The volume is 27 cubic feet. The formula for finding volume is length x width x height. This box would be described as a cube because all three dimensions are the same length.

Do you have students who need extra support in math?
Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor.

Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with your state’s standards and assessments, plus you’ll receive regular reports every step of the way.

Personalized one-on-one math tutoring programs are available for:
– 2nd grade tutoring
– 3rd grade tutoring
– 4th grade tutoring
– 5th grade tutoring
– 6th grade tutoring
– 7th grade tutoring
– 8th grade tutoring

Why not learn more about how it works?