Measuring Triangles

Triangles: Perimeter

Perimeter is a two-dimensional measure of the distance around the figure. For any polygon, the perimeter is simply the sum of the lengths of all of its sides.

Perimeter of a Triangle

The perimeter of this triangle is 5 cm + 6 cm + 7 cm, or 18 cm.

It’s just that easy!

Perimeter is a two-dimensional measure, so it uses units like centimeters, meters, inches, or feet.

Question

What is the perimeter of triangle ABC below?

1. 14.5 in.
2. 15 in
3. 18 in
4. Cannot be determined from the given information

Reveal Answer

Choice B is correct. Since the triangle is isosceles, it has two legs that measure 4 inches each, and a base that measures 7 inches. Therefore, the perimeter is 4 in. + 4 in. + 7 in., or 15 in. Choice A is incorrect, because the segment labeled 3.5 in. is not a side of triangle ABC.

Triangles: Area

The area of a two-dimensional figure is the number of square units it contains. You’ve probably heard of an apartment or house being measured in square feet (ft²). Other examples of square units are square inches (in²) and square centimeters (cm²).

The area of a triangle is given by the formula , where b is the base and h is the height.

It is important to remember that the base and the height must be perpendicular.

Question

What is the area of triangle ABC below?

1. 7 in²
2. 13.125 in²
3. 14 in²
4. 14.5 in²

Reveal Answer

Choice A is correct. The base has a length of 4 in., and the height has a length of 3.5 in., so the area is 7 in². If you answered C, you may have forgotten to multiply the product of the base and height by one-half. If you answered D, you may have calculated the perimeter of the triangle.

Think about why the formula for area contains . We’ll address this in a later section.

Measuring Right Triangles: The Pythagorean Theorem

This is probably the most popular theorem in all of geometry. In fact, it’s pretty important algebraically, as well.

The right triangle below has legs of length a and b, and a hypotenuse of length c.

Legs and Hypotenuse

The Pythagorean Theorem gives the relationship between the lengths of these sides. It says: The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse. (Note: This is only true for right triangles. If the lengths of the sides of any triangle satisfy the Pythagorean Theorem, the triangle must be a right triangle.)

We can take “square” in its algebraic and its geometric senses. Algebraically, the Pythagorean Theorem looks like this:

In the geometric sense, “square” is literally a square and the theorem looks like this:

Pythagorean Theorem

The area of the square with side a is a², the area of the square with side b is ^b2, and the area of the square with side c is c². And the sum of a² and b² is c². That means that the sum of the areas of the two smaller squares is equal to the area of the largest square.

All right, let’s see how to use the theorem. Suppose the two legs of a right triangle measure 3 in. and 4 in. What is the length of the hypotenuse? In the theorem, a and b represent the lengths of the legs, so let a = 3 and b = 4. Now find c:

A 3-4-5 triangle is the most popular Pythagorean triple. A Pythagorean triple is a set of three positive integers that satisfy the Pythagorean Theorem. Multiples of Pythagorean triples are also Pythagorean triples. In other words, since 3-4-5 is a Pythagorean triple, so is 6-8-10 and 9-12-15. Another Pythagorean triple is 5-12-13.

Most, if not all, test questions related to the Pythagorean Theorem involve Pythagorean triples, because they’re easier to compute and they don’t involve irrational numbers (like √2 or 3√5).

Measuring Right Triangles: Trigonometric Functions

Trigonometry literally means “triangle measure.” The trigonometry (or “trig”) that we’ll explore here is restricted to right triangles, so sometimes it’s called right triangle trigonometry.

A trig function is one that relates the lengths of the sides of a right triangle to one of its angle measures. In this lesson, we’ll explore the three basic trig functions: sine, cosine, and tangent.

The sine of an angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse.

Example 1: The base of this right triangle is 10 in. long. What is its height, h?

Example 2: Now let’s find the length of the hypotenuse. We could use the fact that there are 180° in a triangle to find the measure of the other acute angle, or we could simply use the angle we’re given. If we do that, we have an angle and the sides opposite and adjacent to it. We want to find the hypotenuse, so we could use either sine or cosine.

Therefore, the , or about 11.5 in.

Question

Which of the following is the best approximation for leg x in the triangle below? Note that the cos50° is .574, the sin50° is .766, and the tan50° is 1.19.

1. 4 ft.
2. 5.7 ft.
3. 7.9 ft.
4. 12.2 ft.

Reveal Answer

Choice A is the correct answer. We’re given an angle measure and the hypotenuse. We want to find the length of the side adjacent to the given angle, so we need a trig formula that relates the measure of an angle to the adjacent side and to the hypotenuse. The cosine function does that. The value of x is about 4 ft. If you answered B, you may have used the sine function instead of the cosine function.

Review

• The perimeter of a triangle is the sum of the lengths of the sides of the triangle.
• The area of a triangle is given by the formula , where b is the base and h is the height, and the base and height are perpendicular.
• The Pythagorean Theorem relates the lengths of the sides of a right triangle. It states a² + b² = c², where a and b are the lengths of the legs and c is the length of the hypotenuse.
• Trig functions relate the measure of an angle to the sides of a triangle.
• There are three basic trig functions: