Lesson: Chapter - 6

Polygons

Polygons are enclosed geometric shapes that cannot have fewer than three sides. As this definition suggests, triangles are actually a type of polygon, but they are so important on the Math IIC that we gave them their own section. Polygons are named according to the number of sides they have, as you can see in the chart below.

All polygons, no matter the number of sides they possess, share certain characteristics:

• The sum of the interior angles of a polygon with n sides is (n – 2)180^o. So, for example, the sum of the interior angles of an octagon is (8 – 2)180^o = 6 (180^o) = 1080^o.
• The sum of the exterior angles of any polygon is 360^o.
• The perimeter of a polygon is the sum of the lengths of its sides. The perimeter of the hexagon below, for example, is 35.

Regular Polygons

Most of the polygons with more than four sides that you’ll deal with on the Math IIC will be regular polygons—polygons whose sides are all of equal length and whose angles are all congruent (neither of these conditions can exist without the other). Below are diagrams, from left to right, of a regular pentagon, a regular octagon, and a square (also known as a regular quadrilateral):

Area of a Regular Polygon

There is one more characteristic of polygons with which to become familiar. It has to do specifically with regular hexagons. A regular hexagon can be divided into six equilateral triangles, as the figure below shows:

If you know the length of just one side of a regular hexagon, you can use that information to calculate the area of the equilateral triangle that uses the side. To find the area of the hexagon, simply multiply the area of that triangle by 6.

Quadrilaterals

The most frequently seen polygon on the Math IC is the quadrilateral, which is a general term for a four-sided polygon. In fact, there are five types of quadrilaterals that pop up on the test: trapezoids, parallelograms, rectangles, rhombuses, and squares. Each of these five quadrilaterals has special qualities, as shown in the sections below.

Trapezoids

A trapezoid is a quadrilateral with one pair of parallel sides and one pair of nonparallel sides. Below is an example of a trapezoid:

In the trapezoid pictured above, AB is parallel to CD (shown by the arrow marks), whereas AC and BD are not parallel.

where s₁ and s₂ are the lengths of the parallel sides (also called the bases of the trapezoid), and h is the height. In a trapezoid, the height is the perpendicular distance from one base to the other.

Try to find the area of the trapezoid pictured below:

To find the area, draw in the height of the trapezoid so that you create a 45-45-90 triangle. You know that the length of the leg of this triangle—and the height of the trapezoid—is 4. Thus, the area of the trapezoid is ⁶⁺¹⁰/₂ × 4 = 8 × 4 = 32. Check out the figure below, which includes all the information we know about the trapezoid:

Parallelogram

A parallelogram is a quadrilateral whose opposite sides are parallel. The figure below shows an example:

1. Opposite sides are equal.
2. Opposite angles are congruent.
3. Adjacent angles are supplementary (they add up to 180º).

To visualize this last property, simply picture the opposite sides of the parallelogram as parallel lines and one of the other sides as a transversal:

In area problems, you will likely have to find the height using techniques similar to the one used in the previous example problem with trapezoids.

The next three quadrilaterals that we’ll review—rectangles, rhombuses, and squares—are all special types of parallelograms.

Rectangles

A rectangle is a quadrilateral in which the opposite sides are parallel and the interior angles are all right angles. A rectangle is essentially a parallelogram in which the angles are all right angles. Also similar to parallelograms, the opposite sides of a rectangle are equal.

where b is the length of the base, and h is the height.

To find the area of a rhombus, use the same methods as used to find the area of a parallelogram. For example:

A diagonal through the rectangle cuts the rectangle into two equal right triangles. In the figure below, the diagonal BD cuts rectangle ABCD into congruent right triangles ABD and BCD.

Because the diagonal of the rectangle forms right triangles that include the diagonal and two sides of the rectangle, if you know two of these values, you can always calculate the third with the Pythagorean theorem. If you know the side lengths of the rectangle, you can calculate the diagonal; if you know the diagonal and one side length, you can calculate the other side.

Rhombuses

A rhombus is a quadrilateral in which the opposite sides are parallel and the sides are of equal length.

where b is the length of the base, and h is the height.

To find the area of a rhombus, use the same methods as used to find the area of a parallelogram. For example:

If ABCD is a rhombus, AC = 4, and ABD is an equilateral triangle, what is the area of the rhombus?

If ABD is an equilateral triangle, then the length of a side of the rhombus is 4, and angles ADB and ABD are 60º. Draw an altitude from a to DC to create a 30-60-90 triangle, and you can calculate the length of this altitude to be 2 v3. The area of a rhombus is bh, so the area of this rhombus is 4 × 2 v3 = 8 v3 .

Squares

A square is a quadrilateral in which all the sides are equal and all the angles are right angles. A square is a special type of rhombus, rectangle, and parallelogram:

where s is the length of a side of the square. Because all the sides of a square are equal, it is also possible to provide a simple formula for the perimeter: P = 4s, where s is, once again, the length of a side.

From the properties of a 45-45-90 triangle, we know that d=sv2.In other words, if you know the length of one side of the square, you can easily calculate the length of the diagonal. Similarly, if you know the length of the diagonal, you can calculate the length of the sides of the square.

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