Chapter 7 · Measurement and geometry foundations review

Classifying Polygons, Triangles and Quadrilaterals

Learn what makes a figure a polygon and how to name it. Classify triangles and quadrilaterals by their definitions, using stated facts instead of looks.

20 min read

A shape's name is not a matter of opinion. Every name has a definition, and a figure gets the name only when it meets that definition. Start with the widest class in this lesson.

A polygon is a flat (two-dimensional), closed figure whose boundary is made only of straight line segments called sides. The sides meet only at their endpoints, and each meeting point is a vertex (plural: vertices).

Test a figure against every part of that definition:

A circle is closed, but its boundary is curved. It has no straight sides, so it is not a polygon.
Three straight segments joined end to end that do not return to the starting point leave a gap. That figure is not closed, so it is not a polygon.
A closed shape with two straight edges and one curved edge is not a polygon, because part of its boundary is not straight.

Once you know a figure is a polygon, you classify it by counting its sides. The table gives the name for each count from 3 to 10.

Polygon names by number of sides

Number of sides	Name
$3$	Triangle
$4$	Quadrilateral
$5$	Pentagon
$6$	Hexagon
$7$	Heptagon
$8$	Octagon
$9$	Nonagon
$10$	Decagon

As you trace around a polygon, you pass a side, then a vertex, then a side, then a vertex, until you are back at the start. So a polygon has exactly as many vertices as sides, and you can count either one.

The coordinate plane below shows three polygons. The horizontal axis is $x$ , the vertical axis is $y$ , and each grid step is 1 unit. Each polygon is labelled with its number of sides.

The polygon labelled 3 has vertices $(1, 1)$ , $(4, 1)$ and $(2, 5)$ . Three vertices mean three sides, so it is a triangle.
The polygon labelled 5 has vertices $(6, 1)$ , $(8, 1)$ , $(9, 3)$ , $(7, 5)$ and $(5, 3)$ . Five vertices mean five sides, so it is a pentagon.
The polygon labelled 6 has six vertices. You will test and name it in the worked example that follows.

Three polygons, each labelled with its number of sides:

3

5

and

6

Worked example

Polygon or not?

Decide whether each figure is a polygon. If it is, name it. Figure A is closed, and its boundary is one straight edge and one curved arc. Figure B is the figure labelled 6 in the plane above: straight segments join $(10, 1)$ , $(14, 1)$ , $(14, 4)$ , $(12, 4)$ , $(12, 5)$ and $(10, 5)$ in that order, and one more segment joins $(10, 5)$ back to $(10, 1)$ .

Test Figure A against the definition. A polygon's boundary is made only of straight sides. Figure A has a curved arc, so it fails that part of the definition. Figure A is not a polygon, so it gets no polygon name.
Test Figure B for straight sides. Every part of its boundary is a straight segment between two of the listed points, so all of its sides are straight.
Test Figure B for a closed boundary. The last segment returns to the starting point (10, 1), so there is no gap. The segments meet only at the listed points, which are their endpoints. Figure B passes every part of the definition, so it is a polygon.
Count the sides. The six listed points are the vertices, and a polygon has exactly as many sides as vertices. Figure B has 6 sides.
Match the count to its name in the table. A polygon with 6 sides is a hexagon. The unusual shape does not change this, because the name depends only on the number of sides.

Answer

Figure A is not a polygon, because part of its boundary is curved. Figure B is a polygon with $6$ sides, so it is a hexagon.

A triangle is a polygon with three sides, and it has three angles. You can classify a triangle in two separate ways: by its sides and by its angles.

To classify by sides, compare the three side lengths.

A scalene triangle has no two sides equal, such as 4 cm, 6 cm and 7 cm.
An isosceles triangle has at least two equal sides, such as 5 cm, 5 cm and 8 cm.
An equilateral triangle has all three sides equal, such as 7 cm, 7 cm and 7 cm.

The names overlap on purpose. Three equal sides include two equal sides, so every equilateral triangle is also isosceles. The reverse holds only when the third side matches the other two: the triangle with sides 5 cm, 5 cm and 8 cm is isosceles but not equilateral.

To classify by angles, compare each angle measure with $90^\circ$ . The table defines the three angle classes.

Use only stated measures or exact coordinates for these comparisons, never how a drawing looks.

Triangle classes by angles

Class	Definition	Example with stated angle measures
Acute triangle	All three angles measure less than $90^\circ$ .	$50^\circ$ , $60^\circ$ , $70^\circ$
Right triangle	One angle measures exactly $90^\circ$ .	$90^\circ$ , $35^\circ$ , $55^\circ$
Obtuse triangle	One angle measures more than $90^\circ$ .	$110^\circ$ , $40^\circ$ , $30^\circ$

The triangle with vertices

(1, 1)

(5, 1)

and

(1, 4)

. Its horizontal side is

4

units long and its vertical side is

3

units long.

Worked example

Classify a triangle on the coordinate plane

The plane above shows a triangle with vertices $(1, 1)$ , $(5, 1)$ and $(1, 4)$ . The horizontal axis is $x$ , the vertical axis is $y$ , and each grid step is 1 unit. You are told that the slanted side, from $(5, 1)$ to $(1, 4)$ , is $5$ units long. Classify the triangle by its sides and by its angles.

Find the length of the horizontal side. Its endpoints (1, 1) and (5, 1) both have y = 1, so the side is horizontal. Its length is the larger x-coordinate minus the smaller one.
$5 - 1 = 4$
Find the length of the vertical side. Its endpoints (1, 1) and (1, 4) both have x = 1, so the side is vertical. Its length is the larger y-coordinate minus the smaller one.
$4 - 1 = 3$
Compare the three side lengths: 4 units, 3 units and the stated 5 units. No two of them are equal, so by its sides the triangle is scalene.
$3 \ne 4 \qquad 4 \ne 5 \qquad 3 \ne 5$
Find an angle from the coordinates. At (1, 1), the horizontal side meets the vertical side. Horizontal and vertical lines on the coordinate plane are perpendicular, so that angle measures exactly 90°. One angle of exactly 90° makes this a right triangle.

Answer

The triangle is scalene, because its sides are $3$ , $4$ and $5$ units long, and it is right, because its angle at $(1, 1)$ measures $90^\circ$ . It is a scalene right triangle.

Worked example

Classify a triangle from stated measures

A triangle is described without a drawing. Each of its sides is $7$ cm long, and each of its angles measures $60^\circ$ . Give every class name that fits this triangle.

Compare the stated side lengths. All three sides are 7 cm long, so all three are equal and the triangle is equilateral.
$7 = 7 = 7$
Check the isosceles definition, which asks for at least two equal sides. Three equal sides include two equal sides, so the triangle is also isosceles. It is not scalene, because scalene means no two sides are equal.
Compare each stated angle with 90°. Every angle measures 60°, which is less than 90°, so all three angles are less than 90° and the triangle is acute.
$60^\circ < 90^\circ$

Answer

The triangle is equilateral, isosceles and acute.

A quadrilateral is a polygon with four sides. To classify one, you check three kinds of facts: pairs of parallel sides, right angles ( $90^\circ$ ), and sides of equal length. Take those facts from stated measures or exact coordinates, never from how a drawing looks.

Start with parallel sides.

A trapezoid is a quadrilateral with at least one pair of parallel sides.
A parallelogram is a quadrilateral with two pairs of parallel sides.

These classes are nested, like a folder inside a folder. Two pairs of parallel sides include at least one pair, so every parallelogram is also a trapezoid. The reverse holds only when the trapezoid has a second pair of parallel sides.

The table defines three special kinds of parallelogram. Its last column lists the other classes that every figure in that row also belongs to.

Three special kinds of parallelogram

Class	Definition	Also belongs to
Rectangle	A parallelogram with four right angles	Parallelogram, trapezoid
Rhombus	A parallelogram with four sides of equal length	Parallelogram, trapezoid
Square	A parallelogram with four right angles and four sides of equal length	Rectangle, rhombus, parallelogram, trapezoid

Key concept

A figure gets every name whose definition it meets

To classify a figure, compare its stated facts with each definition. Stated facts are side lengths, angle measures in degrees, sides stated to be parallel, and exact coordinates.

A figure belongs to every class whose definition it meets, so one figure can have several names:

A square is also a rectangle and a rhombus.
An equilateral triangle is also isosceles.
Every parallelogram is also a trapezoid.

The nesting goes one way only. A rectangle is a square only when its four sides are also equal in length.

Never classify from how a drawing looks. Sides count as equal or parallel, and an angle counts as a right angle, only when a stated measure, a stated fact or the coordinates show it.

The quadrilateral with vertices

(1, 1)

(6, 1)

(9, 5)

and

(4, 5)

. Its bottom side lies on

y = 1

and its top side lies on

y = 5

Worked example

Find every name that fits a quadrilateral

The plane above shows a quadrilateral with vertices $(1, 1)$ , $(6, 1)$ , $(9, 5)$ and $(4, 5)$ , joined in that order. The horizontal axis is $x$ , the vertical axis is $y$ , and each grid step is 1 unit. You are told that the two slanted sides are parallel and that each one is $5$ units long. Which of these names fit: trapezoid, parallelogram, rectangle, rhombus, square?

Look for parallel sides. The bottom side stays on y = 1 and the top side stays on y = 5, so both are horizontal and they are parallel. The two slanted sides are stated to be parallel. That gives two pairs of parallel sides.
Apply the first two definitions. A trapezoid needs at least one pair of parallel sides, and a parallelogram needs two pairs. This figure has two pairs, so it is a trapezoid and a parallelogram.
Find the lengths of the two horizontal sides. For each one, subtract the smaller x-coordinate from the larger. The bottom side runs from x = 1 to x = 6, and the top side runs from x = 4 to x = 9.
$6 - 1 = 5 \qquad 9 - 4 = 5$
Compare all four side lengths. Each horizontal side is 5 units long, and each slanted side is stated to be 5 units long. A parallelogram with four sides of equal length is a rhombus, so this figure is a rhombus.
$5 = 5 = 5 = 5$
Check for a right angle at (1, 1). The bottom side is horizontal, so a side that meets it at 90° has to be vertical. The side from (1, 1) to (4, 5) is not vertical, because its x-coordinate changes from 1 to 4. The angle at (1, 1) is not 90°.
Apply the last two definitions. A rectangle needs four right angles, and so does a square. This figure has an angle that is not 90°, so it is not a rectangle and not a square.

Answer

The quadrilateral is a trapezoid, a parallelogram and a rhombus. It is not a rectangle or a square, because its angle at $(1, 1)$ is not a right angle.

You classify a figure by testing it against definitions, using stated measures or exact coordinates and never the look of a drawing.

Check that it is a polygon: straight sides, a closed boundary, and sides that meet only at their endpoints. Then count its sides to name it.
For a triangle, compare its side lengths, and compare each angle with $90^\circ$ .
For a quadrilateral, look for parallel pairs, right angles and equal sides.

Keep every name whose definition the figure meets.

Next, you will use these classes to find missing angles in triangles and in quadrilaterals.

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Finding Missing Angles in Triangles and Quadrilaterals

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