### All ACT Math Resources

## Example Questions

### Example Question #1 : How To Find The Area Of A Hexagon

The figure above is a regular hexagon. is the center of the figure. The line drawn is perpendicular to the side.

What is the area of the figure above?

**Possible Answers:**

**Correct answer:**

You can redraw the figure given to notice the little equilateral triangle that is formed within the hexagon. Since a hexagon can have the degrees of its internal rotation divided up evenly, the central angle is degrees. The two angles formed with the sides also are degrees. Thus, you could draw:

Now, the is located on the side that is the same as on your standard triangle. The base of the little triangle formed here is on the standard triangle. Let's call our unknown value .

We know, then, that:

Another way to write is:

Now, there are several ways you could proceed from here. Notice that there are of those little triangles in the hexagon. Since you know that the are of a triangle is:

and for your data...

The area of the whole figure is:

### Example Question #1 : How To Find The Area Of A Hexagon

What is the area of a regular hexagon with a perimeter of ?

**Possible Answers:**

**Correct answer:**

A hexagon has sides. A regular polygon is one that has sides that are of equal length. Therefore, if the side length of our polygon is taken to be , we know:

, or

This question is asking about the area of a regular hexagon that looks like this:

Now, you could proceed by noticing that the hexagon can be divided into little equilateral triangles:

By use of the properties of isosceles and triangles, you could compute that the area of one of these little triangles is:

, where is the side length. Since there are of these triangles, you can multiply this by to get the area of the regular hexagon:

It is likely easiest merely to memorize the aforementioned equation for the area of an equilateral triangle. From this, you can derive the hexagon area equation mentioned above. Using this equation and our data, we know:

### Example Question #1 : How To Find The Area Of A Hexagon

What is the area of a regular hexagon with a side length of miles? Simplify all fractions and square roots in your answer.

**Possible Answers:**

**Correct answer:**

For a hexagon with side length , the formula for the area is

.

We have a side length of 4 miles, so we plug that into the equation and simplify the fraction.

### Example Question #1 : How To Find The Area Of A Hexagon

What is the area of a hexagon with a side of length two? Simplify all fractions and square roots.

**Possible Answers:**

**Correct answer:**

To find the area of a hexagon with a given side length, , use the formula:

Plugging in 2 for and reducing we get:

. (remember order of operations, square first!)

### Example Question #1 : How To Find The Area Of A Hexagon

A single hexagonal cell of a honeycomb is two centimeters in diameter.

What’s the area of the cell to the nearest tenth of a centimeter?

**Possible Answers:**

Cannot be determined

**Correct answer:**

# How do you find the area of a hexagon?

There are several ways to find the area of a hexagon.

- In a regular hexagon, split the figure into triangles.
- Find the area of one triangle.
- Multiply this value by six.

Alternatively, the area can be found by calculating one-half of the side length times the apothem.

## Regular hexagons:

Regular hexagons are interesting polygons. Hexagons are six sided figures and possess the following shape:

In a regular hexagon, all sides equal the same length and all interior angles have the same measure; therefore, we can write the following expression.

One of the easiest methods that can be used to find the area of a polygon is to split the figure into triangles. Let's start by splitting the hexagon into six triangles.

In this figure, the center point, , is equidistant from all of the vertices. As a result, the six dotted lines within the hexagon are the same length. Likewise, all of the triangles within the hexagon are congruent by the side-side-side rule: each of the triangle's share two sides inside the hexagon as well as a base side that makes up the perimeter of the hexagon. In a similar fashion, each of the triangles have the same angles. There are in a circle and the hexagon in our image has separated it into six equal parts; therefore, we can write the following:

We also know the following:

Now, let's look at each of the triangles in the hexagon. We know that each triangle has two two sides that are equal; therefore, each of the base angles of each triangle must be the same. We know that a triangle has and we can solve for the two base angles of each triangle using this information.

Each angle in the triangle equals . We now know that all the triangles are congruent and equilateral: each triangle has three equal side lengths and three equal angles. Now, we can use this vital information to solve for the hexagon's area. If we find the area of one of the triangles, then we can multiply it by six in order to calculate the area of the entire figure. Let's start by analyzing . If we draw, an altitude through the triangle, then we find that we create two triangles.

Let's solve for the length of this triangle. Remember that in triangles, triangles possess side lengths in the following ratio:

Now, we can analyze using the a substitute variable for side length, .

We know the measure of both the base and height of and we can solve for its area.

Now, we need to multiply this by six in order to find the area of the entire hexagon.

We have solved for the area of a regular hexagon with side length, . If we know the side length of a regular hexagon, then we can solve for the area.

If we are not given a regular hexagon, then we an solve for the area of the hexagon by using the side length(i.e. ) and apothem (i.e. ), which is the length of a line drawn from the center of the polygon to the right angle of any side. This is denoted by the variable in the following figure:

## Alternative method:

If we are given the variables and , then we can solve for the area of the hexagon through the following formula:

In this equation, is the area, is the perimeter, and is the apothem. We must calculate the perimeter using the side length and the equation , where is the side length.

## Solution:

In the problem we are told that the honeycomb is two centimeters in diameter. In order to solve the problem we need to divide the diameter by two. This is because the radius of this diameter equals the interior side length of the equilateral triangles in the honeycomb. Lets find the side length of the regular hexagon/honeycomb.

Substitute and solve.

We know the following information.

As a result, we can write the following:

Let's substitute this value into the area formula for a regular hexagon and solve.

Simplify.

Solve.

Round to the nearest tenth of a centimeter.

### Example Question #1 : How To Find If Hexagons Are Similar

Two hexagons are similar. If they have the ratio of 4:5, and the side of the first hexagon is 16, what is the side of the second hexagon?

**Possible Answers:**

**Correct answer:**

This type of problem can be solved through using the formula:

If the ratio of the hexagons is 4:5, then and . denotes area. The side length of the first hexagon is 16.

Substituting in the numbers, the formula looks like:

This quickly begins a problem where the second area can be solved for by rearranging all the given information.

The radical can then by simplified with or without a calculator.

Without a calculator, the 320 can be factored out and simplified:

### Example Question #2 : How To Find If Hexagons Are Similar

The ratio between two similar hexagons is . The side length of the first hexagon is 12 and the second is 17. What must be?

**Possible Answers:**

**Correct answer:**

This kind of problem can be solved for by using the formula:

where the values are the similarity ratio and the values are the side lengths.

In this case, the problem provides , but not . is denoted as in the question.

There are four variables in this formula, and three of them are provided in the problem. This means that we can solve for () by substituting in all known values and rearranging the formula so it's in terms of .

### Example Question #1 : How To Find The Measure Of An Angle

The measure of 3 angles in a triangle are in a 1:2:3 ratio. What is the measure of the middle angle?

**Possible Answers:**

**Correct answer:**60

The angles in a triangle sum to 180 degrees. This makes the middle angle 60 degrees.

### Example Question #1 : How To Find An Angle In A Right Triangle

A 17 ft ladder is propped against a 15 ft wall. What is the degree measurement between the ladder and the ground?

**Possible Answers:**

**Correct answer:**

Since all the answer choices are in trigonometric form, we know we must not necessarily solve for the exact value (although we can do that and calculate each choice to see if it matches). The first step is to determine the length of the ground between the bottom of the ladder and the wall via the Pythagorean Theorem: "x^{2 }+ 15^{2} = 17^{2}"; x = 8. Using trigonometric definitions, we know that "opposite/adjecent = tan(theta)"; since we have both values of the sides (opp = 15 and adj = 8), we can plug into the tangential form tan(theta) = 15/8. However, since we are solving for theta, we must take the *inverse* tangent of the left side, "tan^{-1}". Thus, our final answer is

### Example Question #1 : Triangles

What is the sine of the angle between the base and the hypotenuse of a right triangle with a base of 4 and a height of 3?

**Possible Answers:**

**Correct answer:**

By rule, this is a 3-4-5 right triangle. Sine = (the opposite leg)/(the hypotenuse). This gives us 3/5.