Hero´s formula

Hero's Formula in Math

Definition

Hero's formula, also known as Heron's formula, is a mathematical formula used to find the area of a triangle when the lengths of all three sides are known. It is named after Hero of Alexandria, a Greek mathematician who first discovered and documented this formula.

Knowledge Points

Hero's formula contains the following knowledge points:

1. Understanding the concept of a triangle and its sides.
2. Familiarity with the concept of semiperimeter.
3. Knowledge of the Pythagorean theorem.
4. Understanding the concept of square root.

Formula

The formula for Hero's formula is as follows:

Area = √(s(s-a)(s-b)(s-c))

Where:

• Area represents the area of the triangle.
• s represents the semiperimeter of the triangle.
• a, b, and c represent the lengths of the three sides of the triangle.

Application

To apply Hero's formula, follow these steps:

1. Calculate the semiperimeter (s) of the triangle by adding the lengths of all three sides and dividing the sum by 2.
2. Substitute the values of s, a, b, and c into the formula.
3. Simplify the equation by performing the necessary calculations.
4. Take the square root of the result to find the area of the triangle.

Symbol

The symbol for Hero's formula is not specific to this formula alone. It is represented by the standard mathematical symbols used for area (√) and semiperimeter (s).

Methods

There are various methods to apply Hero's formula, including:

1. Directly substituting the values into the formula.
2. Using a calculator or computer program to perform the calculations.
3. Simplifying the equation by factoring common terms.

Solved Examples

Example 1: Given a triangle with side lengths of 5 cm, 6 cm, and 7 cm, find its area using Hero's formula.

Solution: Using the formula, we calculate the semiperimeter: s = (5 + 6 + 7) / 2 = 9

Substituting the values into the formula: Area = √(9(9-5)(9-6)(9-7)) = √(9 * 4 * 3 * 2) = √(216) ≈ 14.7 cm²

Therefore, the area of the triangle is approximately 14.7 cm².

Example 2: Find the area of a triangle with side lengths of 8 cm, 10 cm, and 12 cm using Hero's formula.

Solution: Calculating the semiperimeter: s = (8 + 10 + 12) / 2 = 15

Substituting the values into the formula: Area = √(15(15-8)(15-10)(15-12)) = √(15 * 7 * 5 * 3) = √(1575) ≈ 39.7 cm²

The area of the triangle is approximately 39.7 cm².

Practice Problems

1. Find the area of a triangle with side lengths of 3 cm, 4 cm, and 5 cm.
2. Calculate the area of a triangle with side lengths of 9 cm, 12 cm, and 15 cm.
3. A triangle has side lengths of 7 cm, 8 cm, and 9 cm. Determine its area.

FAQ

Question: What is Hero's formula? Hero's formula is a mathematical formula used to find the area of a triangle when the lengths of all three sides are known.

Question: Who discovered Hero's formula? Hero's formula is named after Hero of Alexandria, a Greek mathematician who first discovered and documented this formula.