W.3 Identify sides and angles of polygons
What is a polygon?
A polygon is a closed, flat, two-dimensional shape made of straight line segments. The line segments meet at their endpoints and form the complete boundary of the shape. A polygon must be closed, so there cannot be a gap between its sides. Polygons are named according to the number of sides they have.
For example, a triangle has 3 sides, a quadrilateral has 4 sides, a pentagon has 5 sides, a hexagon has 6 sides, a heptagon has 7 sides, and an octagon has 8 sides. Polygons can have different side lengths and different angle measures.
- A triangle is a polygon with 3 sides.
- A rectangle is a polygon with 4 sides.
- A pentagon is a polygon with 5 sides.
- A hexagon is a polygon with 6 sides.
- A shape with a curved edge is not a polygon.
- An open shape with a missing side is not a polygon.
When deciding whether a shape is a polygon, check three things: it must be flat, it must be closed, and all of its sides must be straight line segments.
What are sides of a polygon?
A side of a polygon is one of the straight line segments that forms the boundary of the shape. Each side meets another side at an endpoint. To identify the number of sides, count each line segment around the outside of the polygon exactly once.
The number of sides helps determine the name of a polygon. A polygon with 3 sides is a triangle. A polygon with 4 sides is a quadrilateral. A polygon with 5 sides is a pentagon. A polygon with 6 sides is a hexagon. A polygon with 7 sides is a heptagon. A polygon with 8 sides is an octagon.
- A triangle has 3 sides.
- A quadrilateral has 4 sides.
- A pentagon has 5 sides.
- A hexagon has 6 sides.
- An octagon has 8 sides.
Do not count a corner as a side. A side is the straight segment between two corners. Start at one side and move around the entire shape until you return to where you started.
What are vertices of a polygon?
A vertex is the point where two sides of a polygon meet. The plural form is vertices. Every side has a vertex at each of its endpoints. Because a polygon is closed, the number of vertices is equal to the number of sides.
Vertices are also the locations where the angles of a polygon are formed. For example, a triangle has 3 sides, 3 vertices, and 3 angles. A quadrilateral has 4 sides, 4 vertices, and 4 angles.
- A triangle has 3 sides and 3 vertices.
- A rectangle has 4 sides and 4 vertices.
- A pentagon has 5 sides and 5 vertices.
- A hexagon has 6 sides and 6 vertices.
If you know the number of sides of a polygon, you also know the number of vertices. For a closed polygon, these numbers are always the same.
What are angles of a polygon?
An angle is formed when two line segments meet at a common endpoint. In a polygon, the sides meet at the vertices to form the polygon's angles. The angle is the amount of turn between the two sides.
Angles can have different sizes. In fourth grade, you should be able to recognize and describe acute angles, right angles, and obtuse angles. A right angle measures exactly 90 degrees. An acute angle measures less than 90 degrees. An obtuse angle measures more than 90 degrees but less than 180 degrees.
- An angle that measures 90° is a right angle.
- An angle that measures 45° is an acute angle.
- An angle that measures 120° is an obtuse angle.
- The four corners of a rectangle are right angles.
- A triangle can contain acute angles, right angles, or an obtuse angle, depending on its shape.
The size of an angle is not determined by how long its sides look. A long pair of rays can form a small angle, and a short pair of rays can form a large angle. Look at the amount of opening between the sides.
How to identify acute, right, and obtuse angles
You can identify an angle by comparing it with a right angle. A right angle is exactly 90°. An acute angle is smaller than a right angle, so it measures less than 90°. An obtuse angle is larger than a right angle but does not open as far as a straight angle, so it measures more than 90° and less than 180°.
A square corner can help you recognize a right angle. If the opening of an angle is narrower than a right angle, it is acute. If the opening is wider than a right angle but less than a straight line, it is obtuse.
- 30° is acute because it is less than 90°.
- 90° is right because it is exactly 90°.
- 150° is obtuse because it is greater than 90° and less than 180°.
- If one angle looks like a square corner, check whether it is a right angle.
Remember the order: acute angles are smaller than 90°, right angles are exactly 90°, and obtuse angles are between 90° and 180°. The words describe the size of the angle, not the length of the sides.
Counting sides and angles
Every polygon has the same number of sides, vertices, and angles. To identify these features, trace the outside boundary of the polygon and count each side and each corner once. The corners are the vertices, and the openings at those corners are the angles.
For example, a pentagon has 5 sides. Because each pair of neighboring sides meets at a vertex, the pentagon also has 5 vertices and 5 angles. This relationship works for every polygon.
- A triangle: 3 sides, 3 vertices, and 3 angles.
- A quadrilateral: 4 sides, 4 vertices, and 4 angles.
- A pentagon: 5 sides, 5 vertices, and 5 angles.
- An octagon: 8 sides, 8 vertices, and 8 angles.
If a problem asks for the number of angles in a polygon, you can count the vertices. The number of angles and the number of vertices are the same as the number of sides.
What are equal and unequal sides?
Two or more sides are equal in length when they have the same length. A polygon can have all sides the same length, some sides the same length, or no sides the same length. The lengths of the sides help describe and classify polygons.
A regular polygon has all sides equal in length and all angles equal in measure. A polygon does not have to be regular to be a polygon. For example, a rectangle is generally not a regular polygon because its length and width may be different, even though opposite sides have equal lengths.
- A square has 4 equal sides.
- A rectangle has 2 pairs of equal opposite sides.
- A regular pentagon has 5 equal sides.
- A polygon can still be a polygon even when its sides have different lengths.
Do not assume that sides are equal just because they look equal in a drawing. Use the labels, measurements, or other information provided in the problem.
What are parallel sides?
Parallel lines are lines in the same plane that never meet, even if they are extended forever. In a polygon, two sides are parallel when the sides lie on parallel lines. Parallel sides remain the same distance apart.
Some polygons have parallel sides and some do not. A rectangle and a square each have two pairs of parallel sides. A trapezoid has at least one pair of parallel sides. A triangle has no pair of parallel sides.
- In a rectangle, the top and bottom sides are parallel.
- In a rectangle, the left and right sides are parallel.
- In a square, opposite sides are parallel.
- A triangle has no parallel sides because every pair of sides eventually meets.
Parallel does not mean that two sides have to be the same length. Parallel sides describe their direction and position, not their length.
What are perpendicular sides?
Perpendicular lines are lines that meet to form right angles. When two sides of a polygon meet at a 90° angle, those sides are perpendicular. A corner of a square or rectangle is formed by perpendicular sides.
Perpendicular sides do not have to be the same length. What makes them perpendicular is the angle they form when they meet.
- Adjacent sides of a rectangle are perpendicular because they meet at 90°.
- Adjacent sides of a square are perpendicular because every angle of a square is 90°.
- Two sides that meet at a 45° angle are not perpendicular.
- Two sides that never meet cannot be perpendicular to each other within the polygon.
Use the angle to decide whether sides are perpendicular. If the sides meet to form a right angle, they are perpendicular.
Using sides and angles to describe quadrilaterals
A quadrilateral is a polygon with 4 sides, 4 vertices, and 4 angles. Different quadrilaterals can have different combinations of side and angle characteristics. You can use the number of equal sides, the presence of parallel sides, and the types of angles to describe and classify them.
A square has 4 equal sides, 2 pairs of parallel sides, and 4 right angles. A rectangle has 2 pairs of equal opposite sides, 2 pairs of parallel sides, and 4 right angles. A rhombus has 4 equal sides and 2 pairs of parallel sides, but its angles do not have to be right angles. A trapezoid has at least 1 pair of parallel sides.
- A shape with 4 equal sides and 4 right angles is a square.
- A shape with 4 right angles and opposite sides of equal length is a rectangle.
- A shape with 4 equal sides but angles that are not all right angles can be a rhombus.
- A quadrilateral with at least 1 pair of parallel sides can be classified as a trapezoid.
Do not classify a quadrilateral from only one characteristic. Look at all the information given about its sides, angles, and parallel or perpendicular lines.
How to identify sides and angles in an unfamiliar polygon
When you see a polygon that you do not recognize, you can analyze its parts instead of relying on its name. First, count the sides. Then identify the vertices and angles. Next, examine the angles and determine whether they are acute, right, or obtuse. Finally, look for equal, parallel, or perpendicular sides when the information in the figure allows you to do so.
This process helps you describe a polygon accurately, even when the polygon is turned, enlarged, reduced, or drawn in an unusual way.
- Step 1: Count the straight sides.
- Step 2: Count the vertices where the sides meet.
- Step 3: Identify the angles at the vertices.
- Step 4: Decide whether the angles are acute, right, or obtuse.
- Step 5: Check for equal, parallel, or perpendicular sides if the problem provides enough information.
A polygon can look different when it is rotated. Turning a shape does not change its number of sides, number of angles, or the relationships among its sides. Focus on the geometric properties rather than the direction in which the shape is facing.
Reading geometric information carefully
Geometry problems often provide information through words, measurements, labels, or symbols. Read each part of the figure carefully. A number followed by a degree symbol, such as 90°, describes an angle measure. A number followed by a length unit, such as 6 inches, describes a side length.
Words such as equal, parallel, perpendicular, acute, right, and obtuse give important information about a polygon. Use the information provided rather than making assumptions from the way the picture appears.
- If a problem says two sides are equal, their lengths are the same.
- If a problem says two sides are parallel, the sides do not meet when extended.
- If a problem says two sides are perpendicular, they meet to form a 90° angle.
- If an angle is labeled 90°, you know it is a right angle.
A drawing is a model of the shape, but the written information and measurements are the evidence you should use to solve the problem. Never rely only on how a side or angle looks.
Comparing polygons by their properties
Polygons can be compared by looking at their geometric properties. You can compare the number of sides, the number and types of angles, the lengths of their sides, and whether their sides are parallel or perpendicular. Two polygons may have the same number of sides but have different properties.
For example, both a square and a rectangle have 4 sides and 4 right angles. However, a square has 4 equal sides, while a rectangle has 2 pairs of equal opposite sides. A rhombus also has 4 equal sides, but its angles do not have to be right angles.
- A square and a rectangle both have 4 sides and 4 right angles.
- A square has 4 equal sides, while a rectangle does not have to have 4 equal sides.
- A rectangle and a square both have 2 pairs of parallel sides.
- A triangle and a quadrilateral have different numbers of sides and angles.
When comparing shapes, use complete statements about their properties. For example, say, “Both shapes have 4 sides, but only one has 4 equal sides.” This gives more information than simply saying that the shapes are different.
Important facts to remember
The most important ideas about identifying sides and angles of polygons can be summarized using the properties of the shapes. A polygon is a closed, two-dimensional shape made from straight line segments. Its sides meet at vertices, and angles are formed at those vertices.
The number of sides, vertices, and angles in a polygon is the same. Angles can be acute, right, or obtuse. Sides can be equal or unequal, and some sides may be parallel or perpendicular. These properties help you identify, describe, compare, and classify polygons.
- Triangle: 3 sides, 3 vertices, and 3 angles.
- Quadrilateral: 4 sides, 4 vertices, and 4 angles.
- Pentagon: 5 sides, 5 vertices, and 5 angles.
- Right angle: exactly 90°.
- Acute angle: less than 90°.
- Obtuse angle: greater than 90° but less than 180°.
- Parallel sides: sides on lines that do not meet.
- Perpendicular sides: sides that meet to form right angles.
Before answering a question about a polygon, stop and identify exactly what the question asks. If it asks for sides, count the line segments. If it asks for vertices, count the corners. If it asks for angles, examine the openings at the vertices. If it asks you to classify a shape, use all of the stated properties.
Common Core alignment: CCSS.MATH.CONTENT.4.G.A.2 – Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.
Notes for teachers
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