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Heron's formula allows us to calculate the area of a triangle when the height is unknown, using only the lengths of its sides. It involves calculating the semi-perimeter and subsequently applying the formula to find the area.
In this section, we explore Heron's formula, which calculates the area of a triangle given the lengths of its three sides, denoted as 'a', 'b', and 'c'. The formula is expressed as:
$$ ext{Area} = \sqrt{s(s - a)(s - b)(s - c) } $$ where s is the semi-perimeter given by $$s = \frac{(a + b + c)}{2}$$. This formula is especially useful in cases where the height of the triangle is difficult to determine directly, making it a significant tool in geometrical calculations. We apply this formula through various examples and demonstrate its practicality in real-life scenarios.
Heron's Formula: A method for calculating the area of a triangle given its side lengths.
Semi-Perimeter: The sum of the triangle's sides divided by two, needed for applying Heron's formula.
To find the area, follow this way, / Semi-perimeter first, then area to display.
Imagine a park in the shape of a triangle. You want to plant flowers but need to know how much space you have. Using Heron's formula helps you calculate the area quickly!
SAB: Semi-perimeter, Area, Base dimensions.
{'example': 'Find the area of a triangle with sides 40 m, 32 m, and 24 m.', 'solution': 'Given $a=40$, $b=32$, $c=24$. The semi-perimeter $s = \frac{40 + 32 + 24}{2} = 48 m$. Then: \n$\text{Area} = \sqrt{48(48-40)(48-32)(48-24)} = \sqrt{48(8)(16)(24)} = \sqrt{24576} = 144 m^2.$'}
{'example': 'Find the area of an equilateral triangle with side 10 cm.', 'solution': 'Here, $a = b = c = 10$ cm. Calculate $s = \frac{10 + 10 + 10}{2} = 15$ cm. Then: \n$\text{Area} = \sqrt{15(15-10)(15-10)(15-10)} = \sqrt{15(5)(5)(5)} = \sqrt{375} = 5\sqrt{15} cm^2.$'}
Term: Heron's Formula
Definition: A mathematical formula that calculates the area of a triangle when the lengths of all three sides are known.
A mathematical formula that calculates the area of a triangle when the lengths of all three sides are known.
Term: SemiPerimeter
Definition: Half the perimeter of a triangle, calculated as $$s = \frac{(a + b + c)}{2}$$.
Half the perimeter of a triangle, calculated as $$s = \frac{(a + b + c)}{2}$$.
Term: Area
Definition: The measure of the space enclosed within a polygon.
The measure of the space enclosed within a polygon.
Term: Scalene Triangle
Definition: A triangle with all sides of different lengths.
A triangle with all sides of different lengths.