Chapter 11 · Area of polygons

Classify Polygons by Established Properties

Name polygons by counting sides, then classify triangles and quadrilaterals by properties you establish from given measures or a coordinate grid.

18 min read

You can already plot figures on a coordinate plane and find the lengths of sides that lie on gridlines. Now you will classify figures, which means sorting each one into a named group. You choose the group by checking properties: facts about the figure, such as how many sides it has or which sides are equal. A property counts only when it is established: it is given to you, or you can show it for certain. How a drawing looks is not enough.

A polygon is a flat, closed figure made only of straight sides. Each side meets exactly two other sides, one at each end, and sides never cross. A point where two sides meet is a vertex (plural: vertices). A circle is not a polygon, because its edge is curved. A figure with a gap in its boundary is not a polygon, because it is not closed.

You name a polygon by counting its sides. As you trace around a polygon, each side ends at one vertex, so the number of vertices always equals the number of sides. You can count either one.

Polygon names by number of sides

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

The shape in the next example has corners at

(1, 1)

(7, 1)

(7, 3)

(4, 3)

(4, 6)

and

(1, 6)

Worked example

Identify and name a polygon

The figure shows a shape on a coordinate plane. The $x$ -axis is horizontal, the $y$ -axis is vertical, and gridlines are 1 unit apart. Straight segments join these corners in order and then return to the start: $(1, 1)$ , $(7, 1)$ , $(7, 3)$ , $(4, 3)$ , $(4, 6)$ , $(1, 6)$ . Is the shape a polygon? If it is, what is its name?

Test the definition of a polygon. The boundary is made only of straight segments, and it returns to its starting point, so the figure is closed. Tracing the segments along the gridlines shows that they meet only at the listed corners, so no sides cross. The shape passes every part of the definition, so it is a polygon.
Count the vertices. The list has 6 corners, and each one is a vertex where two sides meet. A polygon has as many sides as vertices, so this polygon has 6 sides.
Match the count to a name in the table. A polygon with 6 sides is a hexagon. The name depends only on the number of sides, so it does not matter that this hexagon is shaped like the letter L.

Answer

The shape is a polygon with 6 sides, so it is a hexagon.

A triangle is a polygon with 3 sides. You can sort triangles in two separate ways: by their side lengths and by their angles.

The size of an angle is measured in degrees, so $90^\circ$ means 90 degrees. A right angle is a square corner, like the corner of a sheet of paper, and it measures exactly $90^\circ$ . An acute angle measures less than $90^\circ$ . An obtuse angle measures more than $90^\circ$ but less than $180^\circ$ .

Every triangle earns exactly one angle name and at least one side name. The table gives the test for each name, with measures that pass it. Side lengths are in centimeters (cm).

Triangle names and the tests they require

Name	Sorted by	Test	Measures that pass the test
Scalene	Sides	No two sides are equal	Sides $5$ cm, $7$ cm, $9$ cm
Isosceles	Sides	At least two sides are equal	Sides $8$ cm, $8$ cm, $3$ cm
Equilateral	Sides	All three sides are equal	Sides $6$ cm, $6$ cm, $6$ cm
Acute	Angles	All three angles are less than $90^\circ$	Angles $50^\circ$ , $60^\circ$ , $70^\circ$
Right	Angles	One angle is exactly $90^\circ$	Angles $90^\circ$ , $35^\circ$ , $55^\circ$
Obtuse	Angles	One angle is greater than $90^\circ$	Angles $110^\circ$ , $40^\circ$ , $30^\circ$

Worked example

Classify a triangle from given measures

A triangle has side lengths $6$ cm, $6$ cm and $6$ cm. Each of its three angles measures $60^\circ$ . Which triangle names does it earn?

Run the side tests, starting with equilateral. All three sides measure 6 cm, so all three sides are equal. The triangle passes the equilateral test.
Run the isosceles test as well. It asks for at least two equal sides. A triangle with three equal sides certainly has at least two, so this triangle is also isosceles. It is not scalene, because the scalene test needs no two sides equal.
Run the angle tests. Each angle measures 60 degrees, and 60 is less than 90, so all three angles are acute. The triangle passes the acute test. No angle is exactly 90 degrees or greater than 90 degrees, so it is not right or obtuse.
$60 < 90$

Answer

The triangle is equilateral, so it is also isosceles, and it is an acute triangle.

A triangle drawn on a coordinate plane gives you two ways to establish properties from the grid itself.

Right angle. Horizontal and vertical gridlines are perpendicular: they cross at right angles. If one side of a triangle lies on a horizontal gridline and another side lies on a vertical gridline, those two sides meet at a right angle.
Equal sides. A mirror line is a line you could fold along so that one half of a figure lands exactly on the other half. Across a vertical gridline, two points are mirror images when they have the same $y$ -coordinate and are the same distance from the line, on opposite sides. Folding does not change any length, so if a fold lands one side exactly on another side, those two sides are equal.

The first triangle has vertices

(1, 1)

(5, 1)

and

(1, 4)

. The second triangle has vertices

(8, 1)

(12, 1)

and

(10, 6)

Worked example

Classify triangles from their coordinates

The figure shows two triangles on a coordinate plane. The $x$ -axis is horizontal, the $y$ -axis is vertical, and gridlines are 1 unit apart. The first triangle has vertices $(1, 1)$ , $(5, 1)$ and $(1, 4)$ . You are told that its side from $(5, 1)$ to $(1, 4)$ is exactly 5 units long. The second triangle has vertices $(8, 1)$ , $(12, 1)$ and $(10, 6)$ . Which names can you establish for each triangle?

Look for sides of the first triangle that lie on gridlines. The side from (1, 1) to (5, 1) lies on the horizontal gridline where y is 1. The side from (1, 1) to (1, 4) lies on the vertical gridline where x is 1. These gridlines are perpendicular, so the angle at (1, 1) is a right angle. The first triangle is a right triangle.
Compare its side lengths. Subtract coordinates along each gridline side: the horizontal side is 4 units long and the vertical side is 3 units long. The third side is given as 5 units long. The lengths 4, 3 and 5 are all different, so no two sides are equal and the first triangle is scalene.
$5 - 1 = 4, \quad 4 - 1 = 3$
Test the vertical gridline where x is 10 as a mirror line for the second triangle. The vertex (10, 6) is on that line. The vertices (8, 1) and (12, 1) have the same y-coordinate. Each is 2 units from the line, on opposite sides, so they are mirror images of each other.
$10 - 8 = 2, \quad 12 - 10 = 2$
Fold along that gridline. The vertex (8, 1) lands on (12, 1), and (10, 6) stays in place. A straight side goes wherever its two ends go, so the side from (8, 1) to (10, 6) lands exactly on the side from (12, 1) to (10, 6). Folding keeps lengths the same, so these two sides are equal. At least two sides are equal, so the second triangle is isosceles.

Answer

The first triangle is a right triangle and a scalene triangle. The second triangle is isosceles. The two grid methods in this lesson do not give it an angle name.

A quadrilateral is a polygon with 4 sides. To classify one, you check for right angles, equal sides and one new property: parallel sides.

Parallel sides lie on lines that never meet and always stay the same distance apart. Sides that share a vertex meet there, so only opposite sides, which share no vertex, can be parallel. A quadrilateral has two pairs of opposite sides to check. On a coordinate grid, two sides that lie on different horizontal gridlines are parallel. So are two sides that lie on different vertical gridlines.

The quadrilateral names form a family, because some tests contain other tests. A figure with two pairs of parallel sides certainly has at least one pair, so a figure that passes the first of those tests also passes the second. The table lists each name, its test and the other names that always come with it.

Quadrilateral names and the tests they require

Name	Test	Other names in this table it always earns
Trapezoid	At least one pair of parallel sides	None
Parallelogram	Two pairs of parallel sides	Trapezoid
Rectangle	Two pairs of parallel sides and four right angles	Parallelogram, trapezoid
Rhombus	Two pairs of parallel sides and four equal sides	Parallelogram, trapezoid
Square	Two pairs of parallel sides, four right angles and four equal sides	Rectangle, rhombus, parallelogram, trapezoid

Key concept

A name is a test, and a figure can pass several

A figure earns a name only when you establish every property that the name requires. Establish each property from given measures or from coordinates on gridlines, never from how the drawing looks.

Some tests contain other tests, so one figure can earn several names. Every equilateral triangle is also isosceles. Every square is also a rectangle and a rhombus. Every parallelogram is also a trapezoid.

The first quadrilateral has vertices

(1, 1)

(5, 1)

(5, 5)

and

(1, 5)

. The second has vertices

(7, 1)

(13, 1)

(10, 4)

and

(7, 4)

Worked example

Give every name a quadrilateral earns

The figure shows two quadrilaterals on a coordinate plane. The $x$ -axis is horizontal, the $y$ -axis is vertical, and gridlines are 1 unit apart. The first quadrilateral has vertices $(1, 1)$ , $(5, 1)$ , $(5, 5)$ and $(1, 5)$ . The second has vertices $(7, 1)$ , $(13, 1)$ , $(10, 4)$ and $(7, 4)$ . Which names from the table does each quadrilateral earn?

Start with the first quadrilateral. Every side lies on a gridline. Its two horizontal sides are parallel, and so are its two vertical sides. At each vertex a horizontal side meets a vertical side, so all four angles are right angles. Each side runs from 1 to 5 along its gridline, so each is 4 units long and all four sides are equal.
$5 - 1 = 4$
Match these facts to the tests. Two pairs of parallel sides make it a parallelogram, and also a trapezoid, because that test asks for at least one pair. Adding four right angles makes it a rectangle. Adding four equal sides makes it a rhombus. Having all three properties makes it a square.
Now take the second quadrilateral. Its bottom side lies on the horizontal gridline where y is 1, and its top side lies on the horizontal gridline where y is 4. These two sides are parallel. One pair of parallel sides passes the trapezoid test, so it is a trapezoid.
Test its other pair of opposite sides. The left side lies on the vertical gridline where x is 7. The fourth side runs from (13, 1) to (10, 4). Its lower end is 6 units from that gridline, and its upper end is only 3 units from it. Parallel sides stay the same distance apart, so these two sides are not parallel. The figure is not a parallelogram.
$13 - 7 = 6, \quad 10 - 7 = 3$
Check the last three tests together. The rectangle, rhombus and square tests each begin with two pairs of parallel sides. This figure has only one parallel pair, so it fails all three tests.

Answer

The first quadrilateral is a square, so it is also a rectangle, a rhombus, a parallelogram and a trapezoid. The second quadrilateral is a trapezoid and earns none of the other four names.

You now have one method for every figure in this lesson: run the tests and keep every name the figure passes. Count sides to name a polygon. Compare side lengths and angle measures to name a triangle. Check parallel sides, right angles and equal sides to name a quadrilateral. Claim only what given measures or gridlines establish, and remember that one figure can earn several names.

Next, you will measure and classify angles.

Up next

Decompose Polygons on a Coordinate Grid

12 min read

Continue