Area Of Non Right Angled Triangles

Here's how to calculate the area of triangles that don't have a right angle, using various methods and formulas.

Area of Non-Right Angled Triangles: A Comprehensive Guide

The area of a triangle represents the amount of space enclosed within its three sides. While the formula ½ * base * height is widely known for right-angled triangles, calculating the area of non-right angled triangles requires different approaches. This guide explores several methods to determine the area of these triangles, providing formulas, explanations, and examples for a thorough understanding.

Why Different Formulas?

The ½ * base * height formula relies on the presence of a right angle, where the height is perpendicular to the base. In non-right angled triangles, this perpendicular height isn't readily available. Therefore, we need alternative methods that utilize the given information, such as side lengths and angles, to accurately calculate the area.

Methods for Calculating the Area

Several methods can be used to find the area of non-right angled triangles, depending on the information available:

1. Using Two Sides and the Included Angle (SAS): This method is used when you know the length of two sides and the measure of the angle between them (the included angle).

2. Using Heron's Formula (SSS): Heron's formula is useful when you know the lengths of all three sides of the triangle.

3. Using the Law of Sines: This method can be employed when you know one side and two angles (AAS or ASA), or two sides and a non-included angle (SSA – with careful consideration of the ambiguous case).

Let's delve into each method in detail.

1. Area Using Two Sides and the Included Angle (SAS)

This method leverages trigonometry to find the area of a triangle when two sides and the included angle are known.

Formula:

Area = ½ * a * b * sin(C)

Where:

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

Explanation:

The formula is derived from the basic area formula (½ * base * height). If we consider side b as the base, the height h can be expressed as a * sin(C) (using trigonometric relationships in a right triangle formed by the height). Substituting this into the basic formula gives us the SAS area formula.

Steps:

1. Identify the two sides and the included angle: Ensure you have the lengths of two sides and the measure of the angle between them.
2. Convert the angle to degrees (if necessary): If the angle is given in radians, convert it to degrees before calculating the sine.
3. Calculate the sine of the angle: Use a calculator or trigonometric table to find the sine of the included angle.
4. Apply the formula: Substitute the values of a, b, and sin(C) into the formula Area = ½ * a * b * sin(C).
5. Calculate the area: Perform the calculation to find the area of the triangle.
6. Include appropriate units: Remember to include the appropriate units for the area (e.g., cm², m², in²).

Example:

Consider a triangle with sides a = 10 cm, b = 12 cm, and the included angle C = 30°.

1. We have a = 10 cm, b = 12 cm, and C = 30°.
2. The angle is already in degrees.
3. sin(30°) = 0.5
4. Area = ½ * 10 cm * 12 cm * 0.5
5. Area = 30 cm²

Therefore, the area of the triangle is 30 square centimeters.

2. Area Using Heron's Formula (SSS)

Heron's formula provides a method to calculate the area of a triangle when the lengths of all three sides are known.

Formula:

Area = √[s(s - a)(s - b)(s - c)]

Where:

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

Explanation:

Heron's formula doesn't rely on angles; it uses only the side lengths to determine the area. It's particularly useful when you can't easily find the height or any angles of the triangle.

Steps:

1. Identify the lengths of the three sides: Ensure you have the lengths of all three sides of the triangle (a, b, c).
2. Calculate the semi-perimeter (s): Use the formula s = (a + b + c) / 2 to find the semi-perimeter.
3. Apply Heron's Formula: Substitute the values of s, a, b, and c into the formula Area = √[s(s - a)(s - b)(s - c)].
4. Calculate the area: Perform the calculation to find the area of the triangle. This often involves taking the square root of a product.
5. Include appropriate units: Remember to include the appropriate units for the area (e.g., cm², m², in²).

Example:

Consider a triangle with sides a = 5 cm, b = 7 cm, and c = 8 cm.

1. We have a = 5 cm, b = 7 cm, and c = 8 cm.
2. Calculate the semi-perimeter: s = (5 + 7 + 8) / 2 = 10 cm
3. Apply Heron's Formula: Area = √[10(10 - 5)(10 - 7)(10 - 8)]
4. Area = √[10 * 5 * 3 * 2] = √300 ≈ 17.32 cm²

Therefore, the area of the triangle is approximately 17.32 square centimeters.

3. Area Using the Law of Sines

The Law of Sines relates the sides of a triangle to the sines of its angles. This can be adapted to calculate the area in certain scenarios.

Formulas:

There are several variations of the Law of Sines area formula, depending on which sides and angles are known:

• Area = (a² * sin(B) * sin(C)) / (2 * sin(A))
• Area = (b² * sin(A) * sin(C)) / (2 * sin(B))
• Area = (c² * sin(A) * sin(B)) / (2 * sin(C))

Where:

• a, b, and c are the lengths of the sides of the triangle.
• A, B, and C are the angles opposite sides a, b, and c, respectively.

When to Use:

• AAS (Angle-Angle-Side): You know two angles and a side that is not between them. You can find the third angle using the fact that the angles in a triangle sum to 180 degrees.
• ASA (Angle-Side-Angle): You know two angles and the side between them. You can find the third angle using the fact that the angles in a triangle sum to 180 degrees.
• SSA (Side-Side-Angle): You know two sides and an angle that is not between them. This is the ambiguous case, and you need to be careful as there might be zero, one, or two possible triangles.

Explanation:

These formulas are derived from the Law of Sines and the SAS area formula. The Law of Sines states that a/sin(A) = b/sin(B) = c/sin(C). By manipulating this relationship and substituting into the SAS formula, we arrive at these area formulas.

Steps:

1. Identify the known information: Determine which sides and angles you know. Make sure you have enough information to use the Law of Sines (AAS, ASA, or SSA).
2. Find any missing angles: If necessary, use the fact that the angles in a triangle sum to 180° to find any missing angles.
3. Choose the appropriate formula: Select the formula that uses the known side and its opposite angle.
4. Calculate the sines of the angles: Find the sines of the angles using a calculator or trigonometric table.
5. Apply the formula: Substitute the values into the chosen formula.
6. Calculate the area: Perform the calculation.
7. Include appropriate units: Remember to include the appropriate units for the area (e.g., cm², m², in²).
8. Address the Ambiguous Case (SSA): If you are given SSA, there may be two possible triangles, one triangle, or no triangles. This needs to be checked carefully.

Dealing with the Ambiguous Case (SSA):

When using the Law of Sines with the SSA case, it's crucial to understand the ambiguous case. Let's say you're given side a, side b, and angle A.

• Case 1: No Triangle: If a < b * sin(A), no triangle can be formed.
• Case 2: One Triangle: If a ≥ b, then there is only one possible triangle.
• Case 3: Two Triangles: If b * sin(A) < a < b, then there are two possible triangles. You'll need to find both possible values for angle B and then solve for the remaining angles and sides for each triangle. The two possible values for angle B are sin⁻¹( (b * sin(A)) / a ) and 180° - sin⁻¹( (b * sin(A)) / a ).

Example (AAS):

Consider a triangle with angle A = 40°, angle B = 60°, and side a = 8 cm.

1. We have A = 40°, B = 60°, and a = 8 cm.
2. Find angle C: C = 180° - A - B = 180° - 40° - 60° = 80°
3. Choose the appropriate formula: Area = (a² * sin(B) * sin(C)) / (2 * sin(A))
4. Calculate the sines: sin(A) = sin(40°) ≈ 0.643, sin(B) = sin(60°) ≈ 0.866, sin(C) = sin(80°) ≈ 0.985
5. Apply the formula: Area = (8² * 0.866 * 0.985) / (2 * 0.643)
6. Area ≈ (64 * 0.866 * 0.985) / 1.286 ≈ 42.47 cm²

Therefore, the area of the triangle is approximately 42.47 square centimeters.

Example (SSA - Careful of Ambiguous Case):

Consider a triangle with side a = 5 cm, side b = 8 cm, and angle A = 30°.

1. We have a = 5 cm, b = 8 cm, and A = 30°.

2. Check for the ambiguous case: b * sin(A) = 8 * sin(30°) = 8 * 0.5 = 4. Since b * sin(A) < a < b (4 < 5 < 8), there are two possible triangles.

3. Find the possible values for angle B:

   • sin(B) = (b * sin(A)) / a = (8 * 0.5) / 5 = 0.8
   • B₁ = sin⁻¹(0.8) ≈ 53.13°
   • B₂ = 180° - sin⁻¹(0.8) ≈ 126.87°

4. Triangle 1: A = 30°, B₁ ≈ 53.13°, C₁ = 180° - 30° - 53.13° ≈ 96.87°

   • Area₁ = (a² * sin(B₁) * sin(C₁)) / (2 * sin(A)) ≈ (5² * sin(53.13°) * sin(96.87°)) / (2 * sin(30°)) ≈ (25 * 0.8 * 0.9925) / 1 ≈ 19.85 cm²

5. Triangle 2: A = 30°, B₂ ≈ 126.87°, C₂ = 180° - 30° - 126.87° ≈ 23.13°

   • Area₂ = (a² * sin(B₂) * sin(C₂)) / (2 * sin(A)) ≈ (5² * sin(126.87°) * sin(23.13°)) / (2 * sin(30°)) ≈ (25 * 0.8 * 0.3939) / 1 ≈ 7.88 cm²

Therefore, there are two possible triangles, with areas approximately 19.85 cm² and 7.88 cm². It's absolutely critical to remember the ambiguous case when using SSA, or you may miss a possible solution.

Other Considerations

• Units: Always pay attention to the units of measurement. The area will be in square units corresponding to the units of the side lengths.
• Accuracy: When using trigonometric functions, use as many decimal places as your calculator allows to minimize rounding errors.
• Calculator Settings: Ensure your calculator is in the correct mode (degrees or radians) for the angle measurements.
• Right-Angled Triangles: While this guide focuses on non-right angled triangles, remember that the basic formula ½ * base * height still applies to right-angled triangles.
• Obtuse Triangles: For obtuse triangles (one angle greater than 90°), the height might fall outside the triangle when using the ½ * base * height approach. You'll need to extend the base and draw the perpendicular height to that extension.

Practical Applications

Calculating the area of non-right angled triangles has numerous practical applications in various fields, including:

• Surveying: Determining land areas and boundaries.
• Navigation: Calculating distances and positions.
• Engineering: Designing structures and calculating material requirements.
• Architecture: Planning building layouts and roof designs.
• Physics: Calculating forces and vectors.
• Computer Graphics: Rendering 3D models.

Advanced Techniques

While the methods described above are sufficient for most cases, more advanced techniques exist for specific scenarios:

• Coordinate Geometry: If the vertices of the triangle are given as coordinates, you can use the determinant formula to calculate the area.
• Vector Cross Product: The magnitude of the cross product of two vectors representing two sides of the triangle is equal to twice the area of the triangle.

Conclusion

Calculating the area of non-right angled triangles requires understanding and applying different formulas based on the available information. The SAS formula, Heron's formula, and the Law of Sines provide effective methods for determining the area in various scenarios. Remember to carefully consider the ambiguous case when using the Law of Sines with SSA. By mastering these techniques, you can accurately calculate the area of any triangle, regardless of its angles. The key is to identify what information is given (sides, angles) and then choose the formula that utilizes that information most effectively. Remember to always include the correct units and be mindful of potential rounding errors.