# ACT Math : Geometry

## Example Questions

### Example Question #1 : Hexagons

If the perimeter of a regular hexagon is , what is the length of a side of that regular hexagon?

Explanation:

Use the formula for perimeter to solve for the length of a side of the regular hexagon:

Where  is perimeter and  is the length of a side.

In this case:

### Example Question #2 : Hexagons

Find the length of one side for a regular hexon with a perimeter of .

Explanation:

Use the formula for perimeter to solve for the side length:

### Example Question #3 : Hexagons

Find the length of one side for a regular hexagon with a perimeter of .

Explanation:

Use the formula for perimeter to solve for the side length:

### Example Question #4 : Hexagons

What is the maximum length of each side for a regular hexago with a perimeter of ?

8.8

Explanation:

Use the formula for perimeter to solve for the side length:

### Example Question #41 : Geometry

What is the side of a Hexagon whose area is ?

Explanation:

To find the side of a hexagon given the area, set the area formula equal to the given area and solve for the side.

### Example Question #41 : Act Math

The sum of all the angles inside of a regular hexagon is . Determine the value of one angle.

Explanation:

In a regular hexagon, all of the sides are the same length, and all of the angles are equivalent. The problem tells us that all of the angles inside the hexagon sum to . To find the value of one angle, we must divide  by , since there are  angles inside of a hexagon.

### Example Question #2 : How To Find An Angle In A Hexagon

All of the angles marked are exterior angles.

What is the value of  in degrees? Round to the nearest hundredth.

Explanation:

There are two key things for a question like this. The first is to know that a polygon has a total degree measure of:

, where  is the number of sides.

Therefore, a hexagon like this one has:

.

Next, you should remember that all of the exterior angles listed are supplementary to their correlative interior angles. This lets you draw the following figure:

Now, you just have to manage your algebra well. You must sum up all of the interior angles and set them equal to . Since there are  angles, you know that the numeric portion will be  or . Thus, you can write:

Simplify and solve for :

This is  or .

### Example Question #3 : How To Find An Angle In A Hexagon

The figure above is a hexagon.  All of the angles listed (except the interior one) are exterior angles to the hexagon's interior angles.

What is the value of ?

Explanation:

There are two key things for a question like this. The first is to know that a polygon has a total degree measure of:

, where  is the number of sides.

Therefore, a hexagon like this one has:

.

Next, you should remember that all of the exterior angles listed are supplementary to their correlative interior angles.  This lets you draw the following figure:

Now, you just have to manage your algebra well. You must sum up all of the interior angles and set them equal to . Thus, you can write:

Solve for :

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

A hexagon is made up of 6 congruent equilateral triangles. Each equilateral triangle has a length of 8 units. What is the area in square units of the hexagon?

Explanation:

First, let's draw out the hexagon.

Because the hexagon is made up of 6 equilateral triangles, to find the area of the hexagon, we will first find the area of each equilateral triangle then multiply it by 6.

Using the Pythagorean Theorem, we find that the height of each equilateral triangle is .

The area of the triangle is then

Multiply this value by 6 to find the area of the hexagon.

### Example Question #2 : How To Find The Area Of A Hexagon

What is the area of a regular hexagon with a side length of ?

Explanation:

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: