Area of SAS & ASA Triangles: Videos & Practice Problems

To calculate the area of any triangle, the fundamental formula is given by:

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

This formula applies universally, including for non-right triangles, as long as the height is known. However, in many cases, the height may not be directly provided. In such scenarios, the height can be determined using the sine of a known angle.

For example, consider a triangle where the base is known, but the height is not explicitly given. To find the height, you can use the sine function from trigonometry. If you have an angle \(A\) and the hypotenuse \(c\), the relationship can be expressed as:

\(\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{c}\)

Rearranging this gives:

\(h = c \times \sin(A)\)

In this case, the area can then be calculated by substituting the height back into the area formula:

Area = \(\frac{1}{2} \times b \times (c \times \sin(A))\)

Thus, the area can also be expressed as:

Area = \(\frac{1}{2} \times b \times c \times \sin(A)\)

It is important to note that this formula can be adapted based on the sides and angles of the triangle. The variations include:

Area = \(\frac{1}{2} \times BC \times \sin(A)\)

Area = \(\frac{1}{2} \times AC \times \sin(B)\)

Area = \(\frac{1}{2} \times AB \times \sin(C)\)

In each case, the angle corresponds to the third vertex of the triangle that is not part of the two sides being multiplied. This approach allows for the calculation of the area of non-right triangles using the relationships between their sides and angles.

To calculate the area of a triangle, the fundamental formula used is:

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

In this scenario, the base \(b\) is given as 8 units. However, determining the height can be challenging, especially in obtuse triangles where the height may extend beyond the base. To find the height, we can utilize the relationship between the sides and angles of the triangle.

We have the following known values: \(a = 6\), \(b = 8\), and the angle \(C = 120^\circ\). The area can also be expressed using the formula:

Area = \(\frac{1}{2} \times a \times b \times \sin(C)\)

In this case, we can rearrange the formula to find the height. The height \(h\) can be expressed as:

Height = \(a \times \sin(C)\)

To find the height, we first need to calculate \(\sin(120^\circ)\). Using the identity that states \(\sin(180^\circ - x) = \sin(x)\), we find:

\(\sin(120^\circ) = \sin(60^\circ)\)

Thus, we can calculate the height as:

Height = \(6 \times \sin(120^\circ) = 6 \times \sin(60^\circ)\)

Calculating \(\sin(60^\circ)\) gives us \(\frac{\sqrt{3}}{2}\), leading to:

Height = \(6 \times \frac{\sqrt{3}}{2} = 3\sqrt{3}\)

Now, substituting the base and height back into the area formula, we have:

Area = \(\frac{1}{2} \times 8 \times 3\sqrt{3}\)

Calculating this gives:

Area = \(12\sqrt{3}\) square units

However, if we approximate \(\sqrt{3} \approx 1.732\), the area can also be expressed numerically as approximately 20.8 square units. This demonstrates that understanding the relationships between the sides and angles of a triangle is crucial for accurately calculating its area, especially in cases involving obtuse angles.

To calculate the area of a triangle when the height is not provided, we can utilize the sine function, especially in cases like the ASA (Angle-Side-Angle) triangle configuration. In this scenario, we have two angles and the side between them, but we need two sides to apply the standard area formulas. However, we can find the missing information using the Law of Sines and the Law of Cosines.

Consider a triangle where angle A is 40 degrees, angle C is 60 degrees, and side b is 4. To find the area, we first need to determine the height of the triangle. The height can be expressed as the product of the hypotenuse (side c) and the sine of angle A. However, since we do not know side c, we must first find it.

Using the Law of Sines, we can set up the equation:

\[ \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \]

Here, we know side b and angles A and C. To find angle B, we apply the angle sum property of triangles:

\[ A + B + C = 180^\circ \Rightarrow B = 180^\circ - 40^\circ - 60^\circ = 80^\circ \]

Now, we can use the Law of Sines again to find side c:

\[ \frac{c}{\sin C} = \frac{b}{\sin B} \Rightarrow c = \frac{b \cdot \sin C}{\sin B} \]

Substituting the known values:

\[ c = \frac{4 \cdot \sin(60^\circ)}{\sin(80^\circ)} \approx 3.5 \]

With side c determined, we can now calculate the height of the triangle. The area A of the triangle can be calculated using the formula:

\[ A = \frac{1}{2} \times b \times h \]

Where h is the height, which can be expressed as:

\[ h = c \cdot \sin A \]

Thus, substituting the known values:

\[ A = \frac{1}{2} \times 4 \times (3.5 \cdot \sin(40^\circ)) \approx 4.5 \]

Therefore, the area of the triangle is approximately 4.5 square units. This method demonstrates how to effectively use trigonometric relationships to find missing sides and ultimately calculate the area of a triangle when certain dimensions are not directly available.