Area of an Oblique Triangle - SAS & SSS - Heron's Formula, Trigonometry
TLDRThis educational video script introduces two methods for calculating the area of a triangle. The first method uses Heron's formula, applicable for a triangle with all three sides known (e.g., sides 7, 6, and 5). It involves calculating the semi-perimeter 's' and then the area through a square root expression involving 's' and the sides. The second method is for oblique triangles with two sides and the included angle known (side-angle-side scenario), using the formula \( \frac{1}{2}ab \sin C \). An example with sides 20 and 25 and a 30-degree angle is provided, resulting in an area of 125 square units. The lesson is a clear guide on finding areas of different types of triangles.
Takeaways
- 📏 The lesson focuses on calculating the area of a triangle when all three sides are known.
- 🔢 To find the area, Heron's formula is used, which requires calculating the semi-perimeter 's' first.
- 📐 's' is calculated as half the perimeter of the triangle, which is (a + b + c) / 2.
- 📉 The area is then found using the formula: √(s(s - a)(s - b)(s - c)).
- 📐 In the example given, sides a, b, and c are 7, 6, and 5 units respectively.
- 🧩 The semi-perimeter 's' for the example is calculated as 9 units.
- 🔢 The area calculation simplifies to √(9 * 2 * 3 * 4), which equals 6√6.
- 📐 The final area of the triangle is approximately 14.7 square units.
- 📐 For a side-angle-side (SAS) triangle, where two sides and the included angle are known, a different formula is used: (1/2)ab * sin(c).
- 📐 The formula can also be written as (1/2)bc * sin(a) or (1/2)ac * sin(b), as long as all three letters are different.
- 📐 An example SAS triangle with side a=20, side b=25, and angle c=30 degrees has an area of 125 square units.
Q & A
What is the main topic of this lesson?
-The main topic of this lesson is calculating the area of a triangle when all three sides are given.
What is the formula used to calculate the area of a triangle when all three sides are known?
-Heron's formula is used to calculate the area of a triangle when all three sides are known.
What is 's' in Heron's formula and how is it calculated?
-In Heron's formula, 's' is the semi-perimeter of the triangle, calculated as half of the triangle's perimeter, which is the sum of all three sides divided by 2.
How do you find the area of a triangle using Heron's formula?
-The area is found by taking the square root of 's' times (s - a) times (s - b) times (s - c), where 'a', 'b', and 'c' are the sides of the triangle.
What is the semi-perimeter 's' for a triangle with sides of 7, 6, and 5 units?
-The semi-perimeter 's' for a triangle with sides of 7, 6, and 5 units is calculated as (7 + 6 + 5) / 2 = 18 / 2 = 9 units.
What is the area of a triangle with sides of 7, 6, and 5 units using Heron's formula?
-The area of the triangle with sides of 7, 6, and 5 units is approximately 14.7 square units, calculated using Heron's formula.
What alternative method can be used to find the area of a triangle if you have two sides and the included angle?
-If you have two sides and the included angle, you can use the formula (1/2) * a * b * sin(c) to find the area of the triangle, where 'a' and 'b' are the sides and 'c' is the included angle.
Can you provide an example of how to use the side-angle-side (SAS) formula to calculate the area of a triangle?
-An example given in the script is a triangle with side 'a' of 20 units, side 'b' of 25 units, and an included angle 'c' of 30 degrees. The area is calculated as (1/2) * 20 * 25 * sin(30), which equals 125 square units.
What are the three different ways to write the side-angle-side (SAS) formula for calculating the area of a triangle?
-The three different ways to write the SAS formula are: (1/2) * a * b * sin(c), (1/2) * b * c * sin(a), and (1/2) * a * c * sin(b).
What is the significance of the sine function in the side-angle-side (SAS) formula?
-The sine function in the SAS formula is used to account for the height of the triangle relative to the base, which is necessary to calculate the area when only two sides and the included angle are known.
How does the script differentiate between an SSS (Side-Side-Side) triangle and a SAS (Side-Angle-Side) triangle?
-The script differentiates between an SSS and a SAS triangle by the information given to calculate the area. For an SSS triangle, all three sides are known, and Heron's formula is used. For a SAS triangle, two sides and the included angle are known, and the SAS formula is used.
Outlines
📐 Calculating the Area of an SSS Triangle Using Heron's Formula
This paragraph introduces the process of calculating the area of a triangle when all three sides are known, using Heron's formula. It begins by assigning sides a, b, and c with lengths 7, 6, and 5 respectively. The first step is to calculate 's', which is half the perimeter of the triangle, calculated as (a + b + c) / 2, resulting in 9 for this example. The area is then found using Heron's formula: √(s(s-a)(s-b)(s-c)). Substituting the values, the formula simplifies to √(9(2)(3)(4)), which further simplifies to √(6 * 9), and then to 3√6. The area is thus 6√6 square units, approximately 14.7 when converted to a decimal. The paragraph also briefly mentions an alternative formula for calculating the area of a triangle when two sides and the included angle are known, which is a = (1/2)ab * sin(c), but emphasizes Heron's formula for SSS (Side-Side-Side) triangles.
📏 Using Side-Angle-Side (SAS) Formula to Calculate Triangle Area
This paragraph explains how to calculate the area of a triangle when you have two sides and the included angle, known as a side-angle-side (SAS) triangle. It presents a formula for this scenario: (1/2)ab * sin(c), where 'a' and 'b' are the sides and 'c' is the included angle. The paragraph provides an example with side lengths of 20 and 25, and an angle of 30 degrees. The calculation begins by taking half of the product of the sides (10 * 25) and then multiplying by the sine of the angle (sin(30°) = 1/2). This results in 5 * 20 * 5 = 500, simplifying to an area of 125 square units. The summary also notes that the formula can be rearranged to accommodate different side-angle combinations as long as all three variables are distinct, ensuring the correct application of the SAS formula.
Mindmap
Keywords
💡Area
💡Triangle
💡Heron's Formula
💡Perimeter
💡Square Root
💡Side Angle Side (SAS)
💡Oblique Triangle
💡Included Angle
💡Sine
💡Decimal
Highlights
Lesson focuses on calculating the area of a triangle given all three sides.
Introduction to Heron's formula for calculating the area of an SSS (Side-Side-Side) triangle.
Explanation of calculating 's', which is half of the triangle's perimeter.
Demonstration of the formula a + b + c / 2 to find 's'.
Use of Heron's formula: √(s(s-a)(s-b)(s-c)) to calculate the area.
Example calculation with sides 7, 6, and 5, resulting in an area of √(9(2)(3)(4)).
Simplification of the area calculation to √(9 * √4 * √6).
Final area calculation result of 6√6, approximately 14.7 square units.
Alternative method for triangles with two sides and the included angle (SAS).
Introduction of the formula 1/2 * a * b * sin(c) for SAS triangles.
Explanation of the flexibility in choosing sides and angles for SAS formula.
Example problem with side lengths 20, 25, and an angle of 30 degrees.
Step-by-step calculation using the SAS formula.
Final area calculation result of 125 square units for the SAS triangle example.
Emphasis on the importance of using the correct sides and angles for the formula.
Summary of the lesson's content and the application of Heron's formula and SAS method.
Transcripts
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