Math Background

Polygons and Solid Figures: Overview

When most people hear the word geometry, they think about shapes. Students should realize they are surrounded by many different kinds of shapes every day. Many of these shapes are two-dimensional plane figures.

Plane figures are flat. They can be closed or not closed.

closed circle, unclosed triangle

Plane figures made up of three or more closed line segments are polygons. Each line segment of a polygon is a side. Polygons are classified by the number of sides.

Triangle (three sides), Quadrilateral (four sides), Pentagon (five sides), Hexagon (six sides), Octagon (eight sides).

A triangle has three sides. There are different kinds of triangles.

  • An equilateral triangle has three sides of equal length.
  • An isosceles triangle has only two sides of equal length.
  • A right triangle has one right angle.
Equilateral triangle, Isosceles triangle, Right triangle

A quadrilateral has four sides. Some quadrilaterals have opposite parallel sides. A parallelogram has two pairs of opposite parallel sides. The sides in each pair are equal in length.


A rectangle is a special kind of parallelogram. It has four right angles.


A square is a special kind of rectangle. All four sides of a square are equal in length.


Students explored rectangles and squares in second grade but may not fully understand the relationship between the two figures. See the second question in “When Students Ask” for a suggestion for reinforcing this concept.

When students understand the concept of plane figures and polygons, they are ready to learn how to find the perimeter and area of some figures.

Perimeter is the measure of the distance around a figure. When you first introduce perimeter, have students find the perimeter of objects in the classroom using non-standard units such as paper clips, footsteps, or handspans. For example, students can find the perimeter of a desk in handspans, the perimeter of a table in shoe lengths, and the perimeter of a mouse pad in paper clips. With this background, students will easily make the transition to using addition to find the perimeter of figures.

quadrilateral with sides measuring four centimeters, two centimeters, two centimeters and two centimeters.

The perimeter is 10 centimeters.

Area is the number of square units that cover a figure without overlapping. Have students trace different objects on square grid paper to estimate area or have them cover different objects in the classroom with square tiles or square pieces of paper.

grid figure

Give your students a variety of hands-on experiences finding perimeter and area such as those offered in the lessons that follow. Discuss with students some situations in their lives when they will need to find perimeter or area; for example, figuring what size rug will fit in a room, how much fencing is needed to enclose a garden, how large a car will fit in a garage, and so on.

Students will use what they learned about polygons to explore solid figures. Unlike plane figures, solid figures are not flat; they have three dimensions.

Some solid figures have curved surfaces; they can roll.

Sphere, Cone, and Cylinder

Notice that the cone and cylinder have both curved and flat surfaces. Flat surfaces are called faces. The faces of the cone and cylinder are circles.

The faces of some solid figures are polygons.

Rectangular Prism, Cube, and Pyramid

Students should understand that they can classify solid figures by their faces. See the lesson “Identifying and Classifying Solid Figures.”

Hands-on activities will help students discover other parts of solid figures. The line segment formed by two faces that meet is an edge. The point where edges meet is a vertex.

Rectangular Prism with edge, face and vertex indicated

Students need frequent practice and reinforcement in identifying the faces, edges, and vertices of different solid figures.

Houghton Mifflin Math Grade 3