### All ACT Math Resources

## Example Questions

### Example Question #202 : Triangles

What is the area of an equilateral triangle with a side length of 5?

**Possible Answers:**

**Correct answer:**

Note that an equilateral triangle has equal sides and equal angles. The question gives us the length of the base, 5, but doesn't tell us the height.

If we split the triangle into two equal triangles, each has a base of 5/2 and a hypotenuse of 5.

Therefore we can use the Pythagorean Theorem to solve for the height:

Now we can find the area of the triangle:

### Example Question #1 : How To Find The Area Of An Equilateral Triangle

What is the area of an equilateral triangle with sides of length ?

**Possible Answers:**

**Correct answer:**

While you can very quickly compute the area of an equilateral triangle by using a shortcut formula, it is best to understand how to analyze a triangle like this for other problems. Let's consider this. The shortcut will be given below.

Recall that from any vertex of an equilateral triangle, you can drop a height that is a bisector of that vertex as well as a bisector of the correlative side. This gives you the following figure:

Notice that the small triangles within the larger triangle are both triangles. Therefore, you can create a ratio to help you find .

The ratio of to is the same as the ratio of to .

As an equation, this is written:

Solving for , we get:

Now, the area of the triangle is merely . For our data, this is: or .

Notice that this is the same as . This is a shortcut formula for the area of equilateral triangles.

### Example Question #2 : How To Find The Area Of An Equilateral Triangle

What is the area of an equilateral triangle with a perimeter of ?

**Possible Answers:**

**Correct answer:**

Since an equilateral triangle is comprised of sides having equal length, we know that each side of this triangle must be or . While you can very quickly compute the area of an equilateral triangle by using a shortcut formula, it is best to understand how to analyze a triangle like this for other problems. Let's consider this. The shortcut will be given below.

Recall that from any vertex of an equilateral triangle, you can drop a height that is a bisector of that vertex as well as a bisector of the correlative side. This gives you the following figure:

Notice that the small triangles within the larger triangle are both triangles. Therefore, you can create a ratio to help you find .

The ratio of to is the same as the ratio of to .

As an equation, this is written:

Solving for , we get:

Now, the area of the triangle is merely . For our data, this is: or .

### Example Question #1 : How To Find The Area Of An Equilateral Triangle

What is the area of a triangle with a circle inscribed inside of it, in terms of the circle's radius R?

**Possible Answers:**

**Correct answer:**

Draw out 3 radii and 3 lines to the corners of each triangle, creating 6 30-60-90 triangles.

See that these 30-60-90 triangles can be used to find side length.

Formula for side of equilateral triangle is

.

Now substitute the new equation that is in terms of R in for S.

### Example Question #2 : How To Find The Area Of An Equilateral Triangle

What is the area of a triangle inscribed in a circle, in terms of the radius R of the circle?

**Possible Answers:**

**Correct answer:**

Draw 3 radii, and then 3 new lines that bisect the radii. You get six 30-60-90 triangles.

These triangles can be used ot find a side length .

Using the formula for the area of an equilateral triangle in terms of its side, we get

### Example Question #203 : Triangles

A circle contains 6 copies of a triangle; each joined to the others at the center of the circle, as well as joined to another triangle on the circle’s circumference.

The circumference of the circle is

What is the area of one of the triangles?

**Possible Answers:**

**Correct answer:**

The radius of the circle is 2, from the equation circumference . Each triangle is the same, and is equilateral, with side length of 2. The area of a triangle

To find the height of this triangle, we must divide it down the centerline, which will make two identical 30-60-90 triangles, each with a base of 1 and a hypotenuse of 2. Since these triangles are both right traingles (they have a 90 degree angle in them), we can use the Pythagorean Theorem to solve their height, which will be identical to the height of the equilateral triangle.

We know that the hypotenuse is 2 so . That's our solution. We know that the base is 1, and if you square 1, you get 1.

Now our formula looks like this: , so we're getting close to finding .

Let's subtract 1 from each side of that equation, in order to make things a bit simpler:

Now let's apply the square root to each side of the equation, in order to change into :

Therefore, the height of our equilateral triangle is

To find the area of our equilateral triangle, we simply have to multiply half the base by the height:

The area of our triangle is

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