Area of any triangle

Area of any Triangle

The Formula for the Area of any Triangle

The area of a triangle is given by the formula:

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

Where $b$ is the base of the triangle and $h$ is the height. Consider the illustration below:

From the figure, we can see where the base and height are located in the triangle.

Applying the Formula to find the Area of any Triangle

Example 1:

The base of a triangle is 12 cm long. If the corresponding height is 7 cm, find the area of the triangle.

Solution:

Consider the figure below:

The area of a triangle is given by $\frac{1}{2} \times b \times h$.

Substitute the given values:

$\text{Area} = \frac{1}{2} \times 12 \, \text{cm} \times 7 \, \text{cm}$

$\text{Area} = 42 \, \text{cm}^2$

Therefore, the area of the triangle is 42 cm².

Example 2:

The lengths of two sides of a triangle are 6 cm and 8 cm. Find the area of the triangle if the included angle is $45^\circ$.

Solution:

Consider the triangle below, name it triangle ABC.

The area of a triangle when two sides and the included angle are known is given by:

$\text{Area} = \frac{1}{2} \times b \times h \times \sin \theta$

Substitute the given values:

$\text{Area} = \frac{1}{2} \times 8 \, \text{cm} \times 6 \, \text{cm} \times \sin 45^\circ$

$\text{Area} = 24 \, \text{cm}^2 \times \sin 45^\circ$

$\text{Area} = 24 \, \text{cm}^2 \times 0.7071 = 16.97 \, \text{cm}^2$

Therefore, the area of triangle ABC is 16.97 cm².