Heron's formula calculator

What is Heron’s formula?

Heron’s formula is a mathematical formula that allows you to find the area of a triangle knowing the lengths of all its sides. It is a powerful tool in geometry that allows finding the area of a triangle without the need to measure its height. The formula is named after the ancient Greek mathematician Heron of Alexandria, who made significant contributions to the development of mathematics and engineering.

Historical background

Heron of Alexandria lived in the 1st century AD and was known for his research in mathematics and mechanics. His works influenced the development of science in medieval Europe and the Middle East. Although Heron’s formula was known before Heron, his treatises led to its widespread dissemination and use.

Application of Heron’s formula

Heron’s formula is widely used in geometry, architecture, and engineering. It saves time and effort when calculating the area of triangles in construction and design when measuring the height of a triangle can be difficult. However, if you need to calculate the area of a triangle knowing other parameters than its three sides, you can use a special triangle area calculator. This tool allows for a quick and accurate calculation of the area based on the parameters you need.

An interesting historical fact about the application of the formula in archaeological excavations is when, during the reconstruction of the ancient city of Dionysopolis, archaeologists stumbled upon building fragments forming triangles with known sides. Using Heron’s formula enabled the accurate determination of the building area without destroying or moving historically valuable artifacts. This helped recreate plans of ancient buildings with high precision.

The formula

Before diving into examples and explanations, let’s study Heron’s formula itself:

$A = \sqrt{p(p-a)(p-b)(p-c)}$

where $A$ is the area of the triangle, $a$, $b$, $c$ are the lengths of the sides of the triangle, and $p$ is the semi-perimeter of the triangle. The semi-perimeter is important as it serves as an intermediate step for simplifying further calculations in the formula, especially when all three sides have different lengths. The semi-perimeter is calculated as:

$p = \frac{a + b + c}{2}$

The advantage of finding the semi-perimeter is that it avoids division within the square root, which would make calculations more complex, especially when working with fractional or irrational numbers.

Examples

Example 1: Equilateral triangle

Consider an equilateral triangle with each side equal to 6.

1. Calculate the semi-perimeter:
   $p = \frac{6 + 6 + 6}{2} = 9$

2. Substitute the values into Heron’s formula:
   $A = \sqrt{9(9-6)(9-6)(9-6)} = \sqrt{9 \times 3 \times 3 \times 3}$

3. Solve:
   $A = \sqrt{243} \approx 15.59$

The area of the triangle is approximately 15.59 square units.

Example 2: Scalene triangle

Imagine a triangle with sides 7, 8, and 9.

1. Calculate the semi-perimeter:
   $p = \frac{7 + 8 + 9}{2} = 12$

2. Substitute into Heron’s formula:
   $A = \sqrt{12(12-7)(12-8)(12-9)} = \sqrt{12 \times 5 \times 4 \times 3}$

3. Solve:
   $A = \sqrt{720} \approx 26.83$

The area of the triangle is approximately 26.83 square units.

Example 3: Right-angled triangle

Suppose we have a right-angled triangle with sides 3, 4, and 5. We know this is a right-angled triangle because $\sqrt{3^2 + 4^2} = 5$.

1. Calculate the semi-perimeter:
   $p = \frac{3 + 4 + 5}{2} = 6$

2. Substitute into Heron’s formula:
   $A = \sqrt{6(6-3)(6-4)(6-5)} = \sqrt{6 \times 3 \times 2 \times 1}$

3. Solve:
   $A = \sqrt{36} = 6$

The area of the triangle is 6 square units, which confirms the known formula for the area of a right-angled triangle ($\frac{1}{2} \times 3 \times 4 = 6$).

Notes

• Heron’s formula is applicable to all types of triangles: acute, obtuse, and right-angled.
• To get correct results, ensure that the sides of the triangle satisfy the triangle inequality: the sum of the two shortest sides must be greater than the length of the longest side.

Frequently asked questions

How to find the area of a triangle if only the lengths of its sides are known?

Use Heron’s Formula. Calculate the semi-perimeter using the lengths of all three sides, then substitute the values into the formula:
$A = \sqrt{p(p-a)(p-b)(p-c)}$

Why is it important to check the triangle inequality when using Heron’s formula?

Checking the triangle inequality ensures that the formula is applied to an actually existing triangle rather than a set of segments that cannot form a triangle.

What to do if one of the sides of the triangle is negative?

The side length of a triangle cannot be negative. It’s necessary to review the initial data.

How does Heron’s Formula work for a right-angled triangle?

For a right-angled triangle, Heron’s Formula gives the same area as the classical formula $\frac{1}{2}ab$ for legs $a$ and $b$, but with a more universal approach.

Heron’s formula and the height of a triangle: what’s the connection?

Calculating the area through height would require first finding the height, which can be challenging in practice. Heron’s formula, on the other hand, allows you to calculate the area without knowing the height, provided all sides are known.

Find the area using Heron’s formula, given the sides of the triangle are 4.5 cm, 6.7 cm, and 8.2 cm.

1. Calculate the semi-perimeter $p$:

$$p = \frac{4.5 + 6.7 + 8.2}{2} = \frac{19.4}{2} = 9.7 \, \text{cm}$$

2. Use Heron’s formula to calculate the area

$$\text{A} = \sqrt{p(p-a)(p-b)(p-c)}$$

Substitute the values:

• $p - a = 9.7 - 4.5 = 5.2 \, \text{cm}$
• $p - b = 9.7 - 6.7 = 3.0 \, \text{cm}$
• $p - c = 9.7 - 8.2 = 1.5 \, \text{cm}$

Now find the area: $$ \text{A} = \sqrt{9.7 \cdot 5.2 \cdot 3.0 \cdot 1.5} \approx \sqrt{226.98} \approx 15.07 , \text{cm}^2

Thus, the area of the triangle with these sides is approximately $$ 15.07 \, \text{cm}^2 $$.