How Do You Derive Heron's Formula for Triangle Area?

TL;DR

Heron's Formula is derived by expressing the area of a triangle in terms of its three side lengths. The proof begins with the standard area formula (1/2 base times height) and involves using the Pythagorean theorem to relate the height and the sides, leading to the area being represented as a function of the semi-perimeter and the side lengths.

Transcript

okay here we'll prove the hijran's formula  and here we go first let's begin with the   triangle and let me just draw it right  here and i will call this to be a b and c   and remember the area of the triangle is  just going to be one half base times height   and let me take this for the base and i would  just say here is going to be the height and... Read More

Key Insights

• 🙃 The Heron's Formula allows for the calculation of a triangle's area using only the lengths of its sides.
• 😒 The proof involves algebraic manipulations and the use of relevant mathematical identities such as the Pythagorean theorem and the difference of two squares.
• 😑 The formula is expressed in terms of the triangle's semi-perimeter, allowing for an efficient calculation of the area.
• 🔺 The formula is widely used in geometry and trigonometry to solve various problems involving triangles.
• ❓ Understanding the derivations and proofs of mathematical formulas enhances mathematical knowledge and problem-solving skills.
• 📛 The Heron's Formula is named after Hero of Alexandria, who first recorded it in his book "Metrica" in the 1st century AD.
• 🏑 The formula is a fundamental tool in fields such as engineering, architecture, and physics, where the area of triangles needs to be determined accurately.

Questions & Answers

Q: How do you prove Heron's Formula?

You begin with the standard area of a triangle, one half base times height, and aim to rewrite the height h in terms of the sides a, b, and c. Dropping an altitude splits the base b into two pieces b1 and b2, forming two right triangles. Applying the Pythagorean theorem to each and combining the equations lets you express h, and therefore the area, using only the three side lengths.

Q: What is the key idea behind the derivation of Heron's Formula?

The key is writing the height h in terms of a, b, and c. As the video says, if you can express h using the sides then you are in good shape, because the area one half base times height then depends only on the side lengths.

Q: How is the Pythagorean theorem used in the proof?

The altitude h divides the base into b1 and b2, creating two right triangles. For the first, b1 squared plus h squared equals a squared; for the second, b2 squared plus h squared equals c squared. Both contain h squared, which is why they can be combined next.

Q: Why do you subtract the two Pythagorean equations?

Subtracting the equations cancels the h squared terms and leaves b1 squared minus b2 squared equals a squared minus c squared. The left side is a difference of two squares, so it factors into (b1 minus b2)(b1 plus b2). Since b1 plus b2 equals b, dividing by b gives b1 minus b2 equal to (a squared minus c squared) over b.

Q: How do you solve for b1 in the proof?

You take b1 minus b2 equals (a squared minus c squared) over b and add it to b1 plus b2 equals b. This cancels b2 and gives 2 b1, and after using a common denominator the result is b1 equals (a squared plus b squared minus c squared) over 2b, all in terms of a, b, and c.

Q: How do you find the height h from b1?

Going back to b1 squared plus h squared equals a squared, you subtract b1 squared to get h squared equals a squared minus b1 squared, then take the positive square root since it is geometry. This gives h as the square root of a squared minus b1 squared, and b1 is already known in terms of a, b, and c.

Q: How is Heron's Formula expressed in terms of the side lengths?

After substituting h back into one half base times height and simplifying, the area is written as the square root of s(s - a)(s - b)(s - c), where s is the semi-perimeter and a, b, and c are the side lengths. The video reaches this by combining fractions over 4b squared and factoring the inside using the difference of two squares.

Q: Why is the difference of two squares used repeatedly in the proof?

It is the main factoring tool of the derivation. It first factors b1 squared minus b2 squared into (b1 minus b2)(b1 plus b2), and later it factors the expression inside the square root, written as (2ab) squared minus (a squared plus b squared minus c squared) squared, which is what collapses the algebra into Heron's Formula.

Summary & Key Takeaways

• The video presents the derivation of the Heron's Formula, which calculates the area of a triangle.

• The derivation starts by considering a triangle and its base and height.

• Several equations and substitutions are made to simplify the formula and express it in terms of the triangle's side lengths.