Area of a Triangle

The formula to calculate the area of any triangle is:

$$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$

• Base ($b$): You can choose any side of the triangle to be the base.

• Height ($h$): This is the perpendicular distance from the chosen base to the opposite vertex.

🌟 Important Tip for Exam:

• You must use the height that matches the base you chose.
• The height must be perpendicular from the base up to the opposite vertex.
• You can’t just pick any side and any height, they must be a pair.

To find the area of any triangle, use this formula:

$$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$

🔎 Example of Finding the Triangle Area:

Let’s find the area of this triangle:

• Base ($b$) = $6\ \text{cm}$

• Height ($h$) = $5\ \text{cm}$

Plug the numbers into the formula:

$$
\begin{align}
\text{Area} &= \frac{1}{2} \times 6\ \text{cm} \times 5\ \text{cm} \\[0.5em]
&= \frac{1}{2} \times 30\ \text{cm}^2 \\[0.5em]
&= 15\ \text{cm}^2
\end{align}
$$

So, the area of the triangle is $15\ \text{cm}^2$.

In a right-angled triangle:

• The base and height are always the two shorter sides that meet at the right angle. They are called the legs.

• The longest side is called the hypotenuse. Do not use the hypotenuse in the area formula.

🔎 Example: Area of a Right-Angled Triangle

Suppose you have a right-angled triangle with sides of 5 cm, 12 cm, and 13 cm. Find its area.

• Step 1: Identify the base and height:

  • The two shorter sides (5 cm and 12 cm) are the legs. 

  • The longest side (13 cm) is the hypotenuse.

• Step 2: Use the two legs in the area formula:

$$
\begin{align}
\text{Area} &= \frac{1}{2} \times 5\ \text{cm} \times 12\ \text{cm} \\[0.5em]
&= \frac{1}{2} \times 60\ \text{cm}^2 \\[0.5em]
&= 30\ \text{cm}^2
\end{align}
$$

So, the area of this right-angled triangle is $30\ \text{cm}^2$.

If you know the area and the base of a triangle, you can find the height using the same formula:

$$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$

This time, you need to rearrange the formula to solve for height.

🔎 Example: Finding the Triangle Height Using Area Formula

If  the area of a triangle is $42\ \text{cm}^2$, and the base is $7\ \text{cm}$. Can you find the height $h$ ?

Set up the formula and solve for $h$:

$$
\begin{align}
\frac{1}{2} \times 7\,\text{cm} \times h &= 42\,\text{cm}^2 \\[0.5em]
7\,\text{cm} \times h &= 42\,\text{cm}^2 \times 2 \\[0.5em]
h &= \frac{84\,\text{cm}^2}{7\,\text{cm}} = 12\,\text{cm}
\end{align}
$$

So, the height of this triangle is $12\ \text{cm}$.