Application to Different Triangles and Quadrilaterals

Heron's formula is an amazing mathematical formula used to find the area of a triangle when you know the lengths of all three sides. Unlike the traditional way of finding the area of a triangle using the base and height, Heron's formula allows you to calculate the area without needing to know the height. This can be especially useful in situations where measuring the height is either difficult or impossible.

Understanding Heron's formula

To apply Heron's formula, you need to know the lengths of the three sides of the triangle. Let's call these sides a, b and c. Heron's formula involves a two-step calculation:

1. First, calculate the semi-perimeter s of the triangle. This is half the sum of the sides of the triangle:

s = (a + b + c) / 2

2. Then, use the semi-perimeter to calculate the area A of the triangle:

A = √(s(s - a)(s - b)(s - c))

Let us simplify and understand this formula with some examples.

Example: Finding the area of a triangle

Consider a triangle with sides a = 7 units, b = 8 units, and c = 5 units. Here's how you would use Heron's formula to find its area:

1. Calculate the semi-perimeter:

s = (7 + 8 + 5) / 2 = 10

2. Plug the values into Heron's formula:

A = √(10(10 - 7)(10 - 8)(10 - 5))

A = √(10 × 3 × 2 × 5)

A = √(300)

A ≈ 17.32 square units

The area of this triangle is approximately 17.32 square units.

Applying Heron's formula to different types of triangles

Although Heron's formula can be used for any triangle, let's see how it applies to different types of triangles, such as equilateral, isosceles, and scalene triangles.

Equilateral triangle

All three sides of an equilateral triangle are equal. Let each side of the equilateral triangle be s. For an equilateral triangle:

• a = b = c = s

The semi-perimeter s will be:

s = (s + s + s) / 2 = 3s/2

The area will be:

A = √((3s/2) * (3s/2 - s) * (3s/2 - s) * (3s/2 - s))

A = √((3s/2) * (s/2) * (s/2) * (s/2))

A = (s²√3)/4

Thus, Heron's formula gives the required result for the area of an equilateral triangle very beautifully.

Isosceles triangle

An isosceles triangle has two equal sides. Let these equal sides be a and the base be b:

• a = a
• B = base

The semi-perimeter s is:

s = (a + a + b) / 2 = (2a + b) / 2

The area is calculated as follows:

A = √((s) * (s - a) * (s - a) * (s - b))

This formula gives you the area with the same steps as before, but note that it reduces easily when a = b, confirming and simplifying the area as it usually should.

Scalene triangle

A scalene triangle is a triangle in which all sides are of different lengths. Heron's formula can be applied directly to scalene triangles without any transformations or reductions, highlighting the flexibility of this method.

• a, b, c - all different lengths

Use the same semi-perimeter and area formulas as before, applying the values directly to find the area:

s = (a + b + c) / 2

A = √(s(s - a)(s - b)(s - c))

Applying Heron's formula to quadrilaterals

You may find this surprising, but Heron's formula can also help with some quadrilaterals. If you can divide a quadrilateral into two triangles, you can use Heron's formula to find the area of those triangles. The simplest way to do this is usually to draw a diagonal.

Example: Finding the area of a quadrilateral

Consider the quadrilateral ABCD. If you draw a diagonal AC, you create two triangles: ABC and ACD. You can find the area of the quadrilateral ABCD by calculating the areas of these two triangles and adding them together.

1. Use Heron's formula to find the area of triangle ABC.
2. Use Heron's formula to find the area of triangle ACD.
3. To find the area of quadrilateral ABCD, add the two areas.

Let us understand this with a numerical example:

Let the lengths of the sides of the quadrilateral be: AB = 8 units, BC = 6 units, CD = 7 units, DA = 5 units, and the length of one diagonal AC is 9 units.

Find the area of triangle ABC:

s1 = (8 + 6 + 9) / 2 = 11.5

Area_ABC = √(11.5(11.5 - 8)(11.5 - 6)(11.5 - 9))

Area_ABC = √(11.5 × 3.5 × 5.5 × 2.5)

Area_ABC = √(553.3125)

Area_ABC ≈ 23.52 square units

Find the area of triangle ACD:

s2 = (9 + 7 + 5) / 2 = 10.5

Area_ACD = √(10.5(10.5 - 9)(10.5 - 7)(10.5 - 5))

Area_ACD = √(10.5 × 1.5 × 3.5 × 5.5)

Area_ACD = √(302.0625)

Area_ACD ≈ 17.38 square units

To find the area of the quadrilateral, add the areas of the two triangles:

Area_Quadrilateral = Area_ABC + Area_ACD

Area_Quadrilateral ≈ 23.52 + 17.38 ≈ 40.9 square units

The area of the quadrilateral ABCD is approximately 40.9 square units.

Conclusion and further applications

Heron's formula is important not only because it provides a simple way to calculate the area of a triangle using only its sides, but it also provides a mathematical technique that can be extended to solve complex problems involving different types of triangles and even specific quadrilaterals. In practice, when dealing with geometric problems, Heron's formula gives you the power to intuitively calculate areas without additional constructions such as heights.

This flexibility and conceptual simplicity allows Heron's formula to remain a staple in mathematics, engineering, physics, and many other fields where geometric calculations are necessary. With further practice and exploration, more elegant solutions to geometry can be created using Heron's principle wherever applicable.

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