We have sent an OTP to your contact. Please enter it below to verify.
Alert
Your message here...
Your notification message here...
For any questions or assistance regarding Customer Support, Sales Inquiries, Technical Support, or General Inquiries, our AI-powered team is here to help!
In this section, we learn how to apply Heron's formula to find the area of various triangles, including an example involving a triangular park. We understand how the semi-perimeter is calculated and its role in determining the area, even for scalene triangles where height is not readily available.
In this section, we explore the application of Heron's formula to find the area of a triangle when the side lengths are known, but the height is not. Heron's formula states that the area of a triangle can be calculated using the semi-perimeter \(s = \frac{a + b + c}{2}\), where \(a, b,\) and \(c\) are the lengths of the sides of the triangle. The area is then given by \[ ext{Area} = \sqrt{s(s - a)(s - b)(s - c)} \]. We apply this formula to a triangular park with sides measuring 40 m, 32 m, and 24 m, leading to a calculated area of 384 m². We also verify this with a right triangle calculation. Through examples and exercises, we solidify our understanding of how to use Heron's formula effectively.
Heron's Formula: A way to calculate the area of a triangle without height.
Semi-perimeter: Important in the calculation of area using Heron's formula.
Verification of area through different methods: Including right triangle calculations.
To find triangle area with sides three, add them and divide by two, you'll see.
Imagine a triangle wondering how to measure its area without height. It discovers Heron’s formula and becomes famous for its versatility.
SAS (semi-perimeter, area, sides) to remember the steps clearly.
{'example': 'Calculate the area of a triangular park with sides 40 m, 32 m, and 24 m.', 'solution': '\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{48(48-40)(48-32)(48-24)} = \sqrt{48(8)(16)(24)} = 384 \text{ m}^2 \]'}
{'example': 'Find the area of an equilateral triangle with each side 10 cm.', 'solution': '\[ s = \frac{10 + 10 + 10}{2} = 15 \text{ cm}, \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{15(15-10)(15-10)(15-10)} = 25\sqrt{3} \text{ cm}^2 \]'}
Term: Heron's Formula
Definition: A formula to calculate the area of a triangle when the lengths of all three sides are known.
A formula to calculate the area of a triangle when the lengths of all three sides are known.
Term: Semiperimeter
Definition: Half the sum of the lengths of the sides of a triangle, used in calculating area via Heron's formula.
Half the sum of the lengths of the sides of a triangle, used in calculating area via Heron's formula.
Term: Scalene Triangle
Definition: A triangle with all sides of different lengths.
A triangle with all sides of different lengths.
Term: Perimeter
Definition: The total length of the sides of a polygon.
The total length of the sides of a polygon.