Unlock Triangle Area: Non-Right Angle Formula Explained!

The Law of Sines, a cornerstone of trigonometry, provides a foundational framework for understanding relationships within non-right triangles. This principle, actively taught in institutions like the Khan Academy, directly informs the application of the area of a non right angle triangle formula. Calculating the area using trigonometry requires understanding the angles and sides of the triangle, often aided by tools such as a scientific calculator, when the height isn't readily available. This formula is incredibly important for different fields that relies on land surveying.

Image taken from the YouTube channel VividMath , from the video titled Area Of A Non-Right Angle Triangle .

Unlocking Triangle Area: The Non-Right Angle Formula Explained

Understanding how to calculate the area of a triangle is a fundamental skill in geometry. While the formula 1/2 base height works perfectly for right-angled triangles, it becomes less convenient when dealing with triangles that lack a readily identifiable right angle and height. This explanation focuses on the "area of a non right angle triangle formula," also known as the Sine Area Formula, and how to effectively apply it.

Identifying the Need for the Non-Right Angle Formula

Often, we are given information about a triangle that doesn't directly provide the base and height. Instead, we might know two sides and the included angle (the angle between those two sides). In these situations, the non-right angle area formula shines.

• When to Use It: You have two sides and the included angle.
• When Not to Use It: You have the base and perpendicular height, or if you know all three sides (Heron's Formula would be more suitable in this case).

The Sine Area Formula: Derivation and Explanation

The area of a non-right-angled triangle formula leverages the sine function to relate the sides and the included angle to the area.

The Formula

The formula is expressed as:

Area = 1/2 a b * sin(C)

Where:

• 'a' and 'b' are the lengths of two sides of the triangle.
• 'C' is the measure of the included angle (the angle between sides 'a' and 'b').
• sin(C) is the sine of angle C.

Derivation

The formula can be derived from the basic area formula (1/2 base height) by expressing the height in terms of a side and the sine of the included angle.

1. Consider a triangle ABC. Let 'a' be the base (BC), and draw an altitude (height 'h') from vertex B to the base AC.
2. Now we have a right-angled triangle where sin(C) = h/b (where 'b' is the length of side AB).
3. Therefore, h = b * sin(C).
4. Substitute this expression for 'h' into the basic area formula: Area = 1/2 a (b * sin(C)).
5. This simplifies to Area = 1/2 a b * sin(C), the Sine Area Formula.

Why does this work?

The sine function relates the angle to the ratio of the opposite side (the height) to the hypotenuse (one of the sides adjacent to the angle) in a right triangle. By using this relationship, we can express the height in terms of the known side and angle, allowing us to calculate the area even without directly knowing the height.

Applying the Formula: Step-by-Step Examples

To effectively use the "area of a non right angle triangle formula", follow these steps:

1. Identify Known Values: Determine which two sides and the included angle are given in the problem.
2. Ensure Correct Units: Make sure the sides are measured in the same units (e.g., centimeters, meters). The angle should be in degrees or radians.
3. Calculate the Sine of the Angle: Use a calculator to find the sine of the included angle. Ensure your calculator is set to the correct angle mode (degrees or radians).
4. Substitute into the Formula: Plug the values of 'a', 'b', and sin(C) into the formula: Area = 1/2 a b * sin(C).
5. Calculate the Area: Perform the calculation to find the area. Remember to include the appropriate units (e.g., cm², m²).

Example 1: Sides and Angle

Let's say we have a triangle with sides a = 8 cm, b = 5 cm, and the included angle C = 60°.

1. Known values: a = 8 cm, b = 5 cm, C = 60°
2. Units: Sides are in cm, angle is in degrees.
3. Calculate sin(60°): sin(60°) ≈ 0.866
4. Substitute: Area = 1/2 8 cm 5 cm * 0.866
5. Calculate: Area ≈ 17.32 cm²

Example 2: Different Angle Units

Suppose we have a triangle with sides a = 10 meters, b = 7 meters, and the included angle C = π/4 radians.

1. Known values: a = 10 m, b = 7 m, C = π/4 radians
2. Units: Sides are in meters, angle is in radians.
3. Calculate sin(π/4): sin(π/4) ≈ 0.707
4. Substitute: Area = 1/2 10 m 7 m * 0.707
5. Calculate: Area ≈ 24.75 m²

Variations and Alternatives

While the basic formula remains the same, its application might slightly vary depending on the given information or problem's context.

Alternate Formula Representations

The formula can be written in different, but equivalent, forms depending on which sides and angle are given:

• Area = 1/2 b c * sin(A)
• Area = 1/2 a c * sin(B)

The key is to always use the two sides and the included angle.

When Heron's Formula is Needed

If you are given all three sides of the triangle, Heron's formula is a better option. Heron's formula is:

Area = √(s(s - a)(s - b)(s - c))

Where:

• a, b, and c are the sides of the triangle.
• s is the semi-perimeter of the triangle, calculated as s = (a + b + c) / 2.

Here's a table to summarize the formula choices based on the information provided:

By understanding the "area of a non right angle triangle formula" and its applications, you can confidently tackle a wide range of triangle area problems.

Video: Unlock Triangle Area: Non-Right Angle Formula Explained!

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So, there you have it! Hopefully, understanding the area of a non right angle triangle formula is a little clearer now. Go forth and conquer those tricky triangle problems!