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Area of a Triangle
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Introduction

In this lesson, you will learn how to calculate the area of a triangle.


This video illustrates the lesson material below. Watching the video is optional.


Area of a Triangle

A triangle is just half of a rectangle, to find the area of a triangle, find the area of the rectangle the triangle fits inside and divide it by 2.

Here are some vocabulary words to help with the lesson.

  • Base: The width of a triangle.
  • Perpendicular: 2 lines that touch at a 90 degree angle.
  • Adjacent Sides: Sides of a shape that meet at a corner.

Before learning about the area of a triangle, consider the rectangle below.
The area of a rectangle is the base multiplied by the height. You can also say it is the length multiplied by the width, or \(A = (l)(w)\).

A rectangle that has a width of 5 and a length of 10. 

Figure 1

In the case above, if the length is 10 and the width is 5, the area of the rectangle is:

\begin{align*} A&=10\times5\\\\ A &=50\end{align*}

What if you had a piece of paper the same shape as the rectangle in Figure 1 and you folded it from the top-left corner to the bottom-right corner? You would see two triangles within the rectangle (see Figure 2). Consider one of these triangles. Where the area of a rectangle is the length times the width, the area of a triangle is:
\begin{align*} A = \frac{1}{2}(l)(w)\end{align*}

A rectangle that has a width of 5 and a length of 10. The rectangle is folded in half diagonally from top left corner to bottom right corner. 

Figure 2

Often, people refer to the length and width of a triangle as the base and height, so the equation is written as:
\begin{align*} A=\frac{1}{2}(b)(h) \end{align*}

Because the equation just involves multiplication, either one of the legs or sides of the triangle could be the base or the height.

This image shows that because the equation involves multiplication, either one of the legs or sides of the triangle could be the base or the height. 

Figure 3

Example 1
A triangle has sides measuring 5 units and 10 units. Calculate the area.

\begin{align*}A&= \frac{1}{2}bh \;\;\;\;\text{or}\;\; \;\;A= \frac{1}{2}lw &\color{red}\small\text{Formula for area of a triangle}\\\\A&= \frac{1}{2}(10)(5) &\color{red}\small\text{Substitute \(b=l=10\) and \(h=w=5\)}\\\\ A&= \frac{1}{2}(50) &\color{red}\small\text{Multiply: \(10\times 5 = 50\)} \\\\ A&= 25 \space units &\color{red}\small\text{Half of 50 is 25}\\\ \end{align*}

The area of the triangle in Figure 3 is 25 units squared.

Conceptually, for this formula to work, the base and the height must be perpendicular to each other. Perpendicular lines are two lines (highlighted gold in Figure 4) that form a 90 degree angle. The right angle symbol () in the corner indicates a 90 degree angle.

This image illustrates that the base and height of a right triangle are perpendicular to each other. For the formula to work, the height and base must be perpendicular to each other. 

Figure 4

Figure 5 shows a triangle where none of the edges are perpendicular to each other. According to the concept of the formula so far in this lesson, for the formula to work, you would need the base and the height to be perpendicular.

However, the following points show the relationship of the triangle in Figure 5 to a rectangle. As you look at each point, you will see that the triangle in Figure 5 is actually half of the area of the rectangle already. This means you can still use the formula to find the area for the triangle in Figure 5.

A triangle that does not have a right angle. 

Figure 5

  • First, make the bottom of the triangle the base of the rectangle.
  • Next, imagine a line that is perpendicular to this base that is only as tall as the vertex, or point, on the other side of the triangle. This will be the height.
  • Now you have a base and a height that are perpendicular to one another.

The bottom of the triangle is the base of the rectangle. You can image a line that is perpendicular to this base that is only as tall as the vertex, or point, on the other side of the triangle. This will be the height. This makes a base and a height that are perpendicular to each other. 

Figure 6

  • If this were a rectangle with the same base and height, it would look something like this:

This figure shows a rectangle made to encompass the triangle from the previous figure. 

Figure 7

  • The whole area of this rectangle is the base multiplied by the height. If you fold these other triangles that are part of the rectangle on the original triangle, you can see that the original triangle is actually half of the entire rectangle.

By folding down the corner triangles that are part of the rectangle around the original triangle, you will see that the original triangle is actually half of the entire rectangle. 

Figure 8

With both corners of the rectangle folded down over the triangle, the triangle shape is emphasized within the rectangle. 

Figure 9

Even though the triangle wasn’t a right triangle, you still calculate it with the formula \(A = \frac{1}{2}(b)(h)\) if the base and height are perpendicular.

Example 2
A triangle has a height of 3 inches and base of 4 inches. Calculate the area of the triangle.

\begin{align*}A&= \frac{1}{2}bh &\color{red}\small\text{Formula for area of a triangle}\\\\A&= \frac{1}{2}(4)(3) &\color{red}\small\text{Substitute given terms}\\\\ A&= \frac{1}{2}(12) &\color{red}\small\text{Multiply} \\\\ A&= 6 \space inches^2 &\color{red}\small\text{Multiply}\\\ \end{align*}

The area of the triangle is \(6in^2\).


Things to Remember


  • The formula for the area of a triangle is \(A =\frac{1}{2}(b)(h)\).
  • Remember that the area of a triangle is the same as the area of a rectangle or square divided by two if they have the same measurements for the length and width.

Practice Problems

  1. A triangle has a base of 10 mm and a height of 12 mm. Use the formula for the area of a triangle to determine the area of this triangle. (
    Solution
    x
    Solution: \(60 \text{ mm}^{2}\)
    )
  2. A triangle has a base of 5 inches and a height of 7 inches. Use the formula for the area of a triangle to determine the area of this triangle. Round to the nearest tenth. (
    Solution
    x
    Solution: \(17.5 \text{ in}^{2}\)
    Details:
    You want to find the area of a triangle with a base of 5 in and a height of 7 in.
    This is a picture of a right triangle with a base that measures 5 in and a height that measures 7 in.
    To find the area of a triangle, it helps to look at a rectangle with sides that are equal to the base and the height that is cut in half in the following way:
    This is a picture of a rectangle with a width of 5 inches and a length of 7 inches. There is a diagonal line from one corner to the opposite corner so that it is cut into two equivalent triangles.
    As you can see, the area of the triangle is half of the area of the rectangle with the same length and width. To find the area of a rectangle, multiply the \(\color{blue} \text{width}\) times the \(\color{red} \text{length}\). To find the area of the triangle, multiply the width times the length then divide it by two.

    \({\text{A}}=\dfrac{1}{2}({\text{b}}\times {\text{h}})\) (Notice how \(\color{blue} \text{width}\) and \(\color{red} \text{length}\) in the triangle formula become \(\color{red} \text{base}\) and \(\color{blue} \text{height}\). These words are used interchangeably.)

    \({\text{A}}=\dfrac{1}{2}(5\times 7)\)

    \({\text{A}}=\dfrac{1}{2}(35)\)

    \({\text{A}}=17.5\)

    The area of the rectangle is \(17.5 \text{ in}^{2}\). Remember to label the area with \(\text{in}^{2}\) since you are measuring area in terms of inches.
    )
  3. A right triangle has perpendicular adjacent sides of lengths 21 cm and 25 cm. Use the formula for the area of a triangle to calculate the area of this triangle. Round to the nearest tenth. (
    Solution
    x
    Solution: \(262.5 \text{ cm}^{2}\)
    )
  4. The top of a slice of blueberry pie is in the shape of a triangle. The slice is 4 inches wide at the widest point and is 7 inches long. Use the formula for the area of a triangle to determine the surface area of the top of this slice of blueberry pie. (
    Video Solution
    x
    Solution: \(14 \text{ in}^{2}\)
    Details:

    (Area of a Triangle #4 (02:08 mins) | Transcript)
    | Transcript)
  5. A garden that is in the shape of a triangle has a width of 35 ft and a length of 55 ft. Use the formula for the area of a triangle to determine the area of this garden. Round to the nearest whole number. (
    Solution
    x
    Solution: \(963 \text{ ft}^{2}\)
    Details:
    You are trying to find the area of a garden shaped like a triangle with a \(\color{blue} \text{width (or base) of 35 ft}\) and a \(\color{red} \text{length (or height) of 55 ft}\).
    This is a picture of a right triangle with a width (or base) of 35 feet, and a length (or height) of 55 feet.
    Remember, the area of a triangle is half the area of a rectangle with the same length and width.
    This is a picture of a rectangle with a width of 35 feet and a length of 55 feet with a diagonal line from one corner to the opposite corner.
    To find the area of a rectangle, multiply the length times the width. To find the area of a triangle, multiply the length times the width and divide it by two. You often refer to the '\(\color{blue} \text{width}\)' and ‘\(\color{red} \text{length}\)’ by the terms '\(\color{red} \text{base}\)’ and '\(\color{blue} \text{height}\)' when talking about triangles.

    In this case, you would multiply 35 by 55, then divide it by 2.

    \({\text{A}}=\dfrac{1}{2}(35\times55)\)

    \({\text{A}}=\dfrac{1}{2}(1925)\)

    \({\text{A}}=962.5\)

    The area of the garden is about \(963 \text{ ft}^{2}\) when rounded to the nearest whole number. And remember, it is labeled \(\text{ft}^{2}\) because you are measuring area in terms of square feet (ft).
    )
  6. A large triangular window has a base of 3 m and a height of 4 m. Use the formula for the area of a triangle to calculate the area of this window. (
    Video Solution
    x
    Solution: \(6 \:{\text{m}}^{2}\)
    Details:

    (Area of a Triangle #6 (01:56 mins) | Transcript)
    | Transcript)

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