Polygons and Triangles Study Guide (page 3)
Introduction to Polygons and Triangles
Geometry existed before the creation.
—PLATO, classical Greek philosopher (427 B.C.E.–347 B.C.E.)
After introducing polygons, this geometry lesson reviews the concepts of area and perimeter and finishes with a detailed exploration of triangles.
We're surrounded by polygons of one sort or another, and sometimes, we even have to do math with them. Furthermore, geometry problems on tests often focus on finding the perimeter or area of polygons, especially triangles. So this lesson introduces polygons and shows you how to work with triangles. The next lesson deals with rectangles, squares, and circles.
What Is a Polygon?
A polygon is a closed, planar (flat) figure formed by three or more connected line segments that don't cross each other. Familiarize yourself with the following polygons; they are the three most common polygons appearing on tests —and in life.
Perimeter of Polygons
Perimeter is the distance around a polygon. The word perimeter is derived from peri, which means around (as in periscope and peripheral vision), and meter, which means measure. Thus, perimeter is the measure around something. There are many everyday applications of perimeter. For instance, a carpenter measures the perimeter of a room to determine how many feet of ceiling molding she needs. A farmer measures the perimeter of a field to determine how many feet of fencing he needs to surround it.
Perimeter is measured in length units, like feet, yards, inches, meters, and so on. To find the permineter of a polygon, add the lengths of the sides.
Example: Find the perimeter of the polygon below:
Solution: Write down the length of each side and add:
The notion of perimeter also applies to a circle; however, the perimeter of a circle is referred to as its circumference.
Area of Polygons
Area is the amount of space taken by a figure's surface. Area is measured in square units. For instance, a square that is 1 unit on all sides covers 1 square unit. If the unit of measurement for each side is feet, for example, then the area is measured in square feet; other possibilities are units like square inches, square miles, square meters, and so on.
You could measure the area of any figure by counting the number of square units the figure occupies. These first two figures below are easy to measure because the square units fit into them evenly, while the next two figures are more difficult to measure because the square units don't fit into them evenly.
Because it's not always practical to measure a particular figure's area by counting the number of square units it occupies, an area formula is used. As each figure is discussed, you'll learn its area formula. Although there are perimeter formulas as well, you don't really need them (except for circles) if you understand the perimeter concept: It is merely the sum of the lengths of the sides.
What Is a Triangle?
A triangle is a polygon with three sides, like those shown here:
The symbol used to indicate a triangle is . Each vertex—the point at which two lines meet—is named by a capital letter. The triangle is named by the three letters at the vertices, usually in alphabetical order: ABC.
There are two ways to refer to a side of a triangle:
- By the letters at each end of the side: AB
- By the letter—typically a lowercase letter—next to the side:
Notice that the name of the side is the same as the name of the angle opposite it, except the angle's name is a capital letter.
There are two ways to refer to an angle of a triangle:
- By the letter at the vertex:A
- By the triangle's three letters, with that angle's vertex letter in the middle:BAC or CAB
The sum of any two sides of a triangle must always be greater than the third side. Therefore, the following could not be the lengths of the sides of a triangle: 4-7-12.
Types of Triangles
A triangle can be classified by the size of its angles and sides:
- 3 congruent angles, each 60°
- 3 congruent sides
Hook: The word equilateral comes from equi, meaning equal, and lat, meaning side. Thus, all equal sides.
- 2 congruent angles, called base angles; the third angle is the vertex angle.
- Sides opposite the base angles are congruent.
- An equilateral triangle is also isosceles.
Hook: Think of the "I" sound in isosceles as two equal eyes, which almost rhymes with 2 equal sides.
- No congruent sides
- No congruent angles
Hook: Scalene, like scaly, could describe a hideous monster or something else as uneven as a triangle where every side and every angle is different.
- 1 right (90°) angle, the largest angle in the triangle
- Side opposite the right angle is the hypotenuse, the longest side of the triangle. (Hook: The word hypotenuse reminds us of hippopotamus, a very large animal.)
- The other two sides are called legs.
- A right triangle may be isosceles or scalene.
Area of a Triangle
To find the area of a triangle, use this formula:
Although any side of a triangle may be called its base, it's often easiest to use the side on the bottom. To use another side, rotate the page and view the triangle from another perspective.
A triangle's height (or altitude) is represented by a perpendicular line drawn from the angle opposite the base to the base. Depending on the triangle, the height may be inside, outside, or on the legs of the triangle. Notice the height of the second triangle: We extended the base to draw the height perpendicular to the base. The third triangle is a right triangle: One leg may be its base and the other its height.
Hook: Think of a triangle as being half a rectangle. The area of that triangle (as well as the area of the largest triangle that fits inside a rectangle) is half the area of the rectangle.
Example: Find the area of a triangle with a 2-inch base and a 3-inch height.
- Draw the triangle as close to scale as you can.
- Label the size of the base and height.
- Write the area formula; then substitute the base and height numbers into it:
- The area of the triangle is 3 square inches.
The following rules tend to appear more frequently on tests than other rules. A typical test question follows each rule.
Example: One base angle of an isosceles triangle is 30°. Find the vertex angle.
- Draw a picture of an isosceles triangle. Drawing it to scale helps: Since it is an isosceles triangle, draw both base angles the same size (as close to 30° as you can), and make sure the sides opposite them are the same length. Label one base angle as 30°.
- Since the base angles are congruent, label the other base angle as 30°.
- There are two steps needed to find the vertex angle:
- Add the two base angles together: 30°+ 30°= 60°
- The sum of all three angles is 180°. To find the vertex angle, subtract the sum of the two base angles (60°) from 180: 180°–60° = 120°
- Thus, the vertex angle is 120°.
Check: Add all 3 angles together to make sure their sum is 180°: 30°+ 30°+ 120°=180°
The longest side of a triangle is always opposite the largest angle, and the smallest side is opposite the smallest angle. That is why in an isosceles triangle, the two base angles are equal to and opposite of the two equal sides.
This rule implies that the second longest side is opposite the second largest angle, and the shortest side is opposite the smallest angle.
Visualize a door and its hinge. The more the hinge is open (largest angle), the fatter the person who can get through (longest side is opposite); similarly, for a door that's hardly open at all (smallest angle), only a very skinny person can get through (shortest side is opposite).
Example: In the triangle shown at the right, which side is the shortest?
- Determine the size of A, the missing angle, by adding the two known angles, and then subtracting their sum from 180°: Thus, A is 44°.
- Since A is the smallest angle, side a (opposite A) is the shortest side.
Right triangles have a rule of their own. Using the Pythagorean theorem, we can calculate the missing side of a RIGHT triangle. a2+ b2 = c2(c refers to the hypotenuse)
Example: What is the perimeter of the triangle shown at the right?
- Since the perimeter is the sum of the lengths of the sides, we must first find the missing side. Use the Pythagorean theorem:
- Substitute the given sides for two of the letters. Remember: Side c is always the hypotenuse:
- To solve this equation, subtract 9 from both sides:
Then, take the square root of both sides. (Note: Refer to Lesson 20 to learn about square roots.) Thus, the missing side has a length of 4 units.
- Adding the three sides yields a perimeter of 12:
Sometime today you'll be bored, and doodling is a good way to pass the time. Doodle with purpose: Draw a triangle! After you draw it, examine it closely to determine if it's an equilateral, isosceles, scalene, or right become! Practice with these shapes until you know them all by heart.
Find practice problems and solutions for these concepts at Polygons and Triangles Practice Questions.
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