Finding the Area of a Right-Angled Triangle

In this tutorial, we will learn how to calculate the area of a right-angled triangle when the base and an angle are known. This method uses trigonometry to find the missing height and then calculate the area.

Formula

The area of a right-angled triangle given the base and an angle is calculated using the formula:

\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \]

If the height is not directly known, it can be calculated using trigonometric functions. For example, if the angle \(\theta\) and the base \(b\) are known, the height \(h\) can be found as:

\[ h = b \cdot \tan(\theta) \]

Explanation of Terms:

  • Base: The known side adjacent to the angle \(\theta\).
  • Height: The perpendicular side opposite the angle \(\theta\).
  • \(\theta\): The angle between the base and the hypotenuse.

Examples

Example 1: Find the area of a right-angled triangle with base = 5 units and angle = 30°.

Given: Base = 5, \(\theta = 30^\circ\)

First, calculate the height using \(h = b \cdot \tan(\theta)\):

\[ h = 5 \cdot \tan(30^\circ) \]

Using \(\tan(30^\circ) = \frac{\sqrt{3}}{3} \approx 0.577\):

\[ h = 5 \cdot 0.577 = 2.885 \]

Now substitute into the area formula:

\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \] \[ \text{Area} = \frac{1}{2} \times 5 \times 2.885 \]

Simplify:

\[ \text{Area} = \frac{1}{2} \times 14.425 = 7.2125 \]

Answer: The area of the triangle is approximately 7.21 square units.

Example 2: Find the area of a right-angled triangle with base = 8 units and angle = 45°.

Given: Base = 8, \(\theta = 45^\circ\)

First, calculate the height using \(h = b \cdot \tan(\theta)\):

\[ h = 8 \cdot \tan(45^\circ) \]

Using \(\tan(45^\circ) = 1\):

\[ h = 8 \cdot 1 = 8 \]

Now substitute into the area formula:

\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \] \[ \text{Area} = \frac{1}{2} \times 8 \times 8 \]

Simplify:

\[ \text{Area} = \frac{1}{2} \times 64 = 32 \]

Answer: The area of the triangle is 32 square units.

This method allows you to calculate the area of a right-angled triangle when the base and an angle are known, using trigonometric principles!