In the realm of geometry, the area of a right triangle serves as a captivating subject that deepens our understanding of the intricate world of shapes. From fundamental theorems to intricate applications, join me as we embark on a journey to unravel the mysteries and logic behind right triangles in this enlightening piece. Let the learning commence!
1. Definition and Fundamental Properties of Right Triangle.
A right triangle is a type of triangle that has one angle equal to 90 degrees. Consider a right triangle ΔABC with a right angle ∠B as shown below, we have:
 The two sides AB and BC are called the legs (also known as the perpendicular sides), often denoted as a and b.
 The side opposite to the right angle ∠B is called the hypotenuse, commonly denoted as c. This is also always the longest side of the right triangle.
 The two acute angles ∠A and ∠C are complementary (which means the sum of these two angles will be 90 degrees).
 If we consider one side of a right angle as the base of a triangle, the other side of the right angle can be seen as the corresponding height of that base (according to the definition of the altitude of a triangle).
 Pythagorean theorem for right triangles: . This theorem is named after the renowned philosopher and mathematician Pythagoras of ancient Greece, discussing the fundamental relationship in Euclidean geometry between the three sides of a right triangle, and is stated as “The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.”
2. Formulas to calculate the area of a right triangle
Method 1: Basic Formula
Apply to the example triangle, we have: S = ½ * c * h = ½ * 11.7 * 5.1 ≈ 30 (cm^{2}).
Explanation:

If we copy a triangle and rotate the copy by 180 degrees, then combining these two triangles will result in a parallelogram. The base and height of this parallelogram will exactly match the hypotenuse and height of the original right triangle. Since the area of a parallelogram is given by the formula S = c * h, it can be concluded that the area of the triangle is equal to half of the product of c and h, or S = (c * h) / 2.
 The formula is still commonly referred to as the “Basic Area Formula for Triangles”. In 300 BC, the ancient Greek mathematician Euclid – the “father of Geometry” – proved this in his book “Elements”.
Formula 2: Pythagorean Theorem
Applying to the example triangle, we have: S = ½ * a * b = ½ * 6 * 10 = 30 (cm^{2}).
Explanation:
 Also the operation to join triangles as in method 1 but with two right triangles, we will have a rectangle with the length and width being the two legs. Because the area of the rectangle is S = a * b, it follows that the area of the triangle is S = (a * b) / 2.
 It is believed that Pythagoras, while waiting to deliver square paving stones to King Polycrates in a palace on the island of Samos, ancient Greece, noticed that if a square tile was cut in half diagonally, the two resulting triangular pieces would be equal in area to half of the original square tile.
However, actual historical records of mathematical knowledge from ancient civilizations such as Liangzhu in ancient China or Egypt have proven that Pythagoras was not the first person to discover this area formula.
Formula 3: Heron’s Method
With S being the area of the triangle, s being half of the triangle’s perimeter (s = (a + b + c) / 2), and a, b, c being the lengths of the triangle’s three sides.
Applying to the example triangle, we have: S = sqrt(13.85 * (13.85 – 6) * (13.85 – 10) * (13.85 – 11.7)) ≈ 30 (cm^{2}).
Explanation:
 Consider the basic geometric formula for the area of a triangle with a as the hypotenuse and h as the corresponding height, we have: S = (a * h_{a}) / 2. From here, we can deduce that h_{a} = (2 * S) / A. Comparing these two formulas, we have:
 This is one of the easiest ways to prove the Heron’s formula – named after the ancient Greek mathematician, engineer Hero of Alexandria, who lived in the 1st century AD. He was known as the “encyclopedia of mathematics” by his contemporaries. Among Hero’s notable works is the bookMetrica (Measurements) – a comprehensive collection of knowledge on geometry and measurement, which includes this formula. However, it is believed that Archimedes, the greatest mathematician in history, knew this formula two centuries earlier.
Method 4: Trigonometric Formula
With ab, bc, ac being the adjacent sides and A, B, C being the three angles of the triangle.
You don’t need to use this formula for the right angle and the two adjacent sides of the right angle because the value of the sine of a 90degree angle is 1.
Applying to the example triangle, we have:
S = ½ * a * c * sin(A) = ½ * 6 * 11.7 * sin(59^{o}) ≈ 30 (cm^{2}).
Explanation:
 Using the definition of sine for a triangle, we have:sin(C) = h / b. Therefore:h = b * sin(C). Substituting the height in the basic triangle area formula, we have:S = ( a * b * sin (C) ) / 2.
 One of the pioneering names in the development of trigonometric formulas, including the formula for computing the area of a geometric figure using trigonometry, is Brahmagupta (598–668 CE) with his work “Brahmasphutasiddhanta.”
Formula 5: The method to solve for the opposite side using adjacent side and hypotenuse
where a and b are the lengths of the two sides of the right triangle; A, C are the measures of the two acute angles corresponding to the two legs of the right triangle.
Applying to example triangle, we have:S = ½ * a^{2} * tan(A) = ½ * 62 * tan(59^{o}) ≈ 30 (cm^{2}).
This formula is derived by Pythagoras based on one of his right triangle theorems: “In any right triangle, each side of the right angle of the triangle is equal to … the other side of the right angle of the triangle multiplied by the tangent of the angle opposite or multiplied by the cotangent of the adjacent angle”.
Formula 6: Inscribed Circle Radius Technique
where p is the semiperimeter of the right triangle and r is the radius of the inscribed circle of that right triangle.
For example, draw an incircle for the given triangle ABC:
Applying to the given triangle example, we have:S = (P * r) / 2 = ((6 + 10 + 11.7) * 2.2) / 2 ≈ 30 (cm^{2}).
This formula is also known as the “Brahmagupta formula,” which is also found in the book Brahmasphutasiddhanta.
Formula 7: The circumference of a circumscribed circle
In this equation, R represents the radius of the circumcircle of the rightangled triangle; a, b, c are the lengths of the three sides of the rightangled triangle.
For example: Draw the circumcircle for the sample triangle ABC:
Applying to the example triangle, we have: S = (a * b * c) / (4 * R) = (6 * 10 * 11.7) / (4 * 5.85) = 702 / 23.4 = 30 (cm^{2}).
The formula is essentially an application of Ptolemy’s theorem: “In a cyclic quadrilateral, the product of the lengths of the diagonals is equal to the sum of the products of the lengths of the opposite sides,” but applied to a circumcircle triangle. This theorem was discovered by the ancient Greek scholar Claudius Ptolemy (100 – 170 AD).
Furthermore, there is another way to apply the circumradius to calculating the area of a right triangle.
where R is the radius of the circumcircle of the right triangle; A and B are the two acute angles of that right triangle.
The rephrased content is:
This formula is a variant of applying the sine law in a triangle with 3 angles A, B, C, but it has been simplified by excluding the right angle C since the sine of a 90degree angle is equal to 1. The original formula with all 3 angles expressed in detail was first presented in the book Brahmasphutasiddhanta.
Formula for an isosceles right triangle
With a as the length of the square’s side.
Example: Consider a right isosceles triangle ABC as follows:
With the rightangled isosceles triangle ABC as shown in the figure above, we have a = 3 cm. Applying the formula, we have:
The formula for calculating the area of a square is S = ½ * a^{2} = ½ * 32 = 4.5 (cm^{2}).
In an equilateral right triangle, the two right angles have the same length. Applying this to the Pythagorean theorem formula, we will have the above formula.
Collection of Formulas for Calculating Area of a Plane Triangle
In addition, in 1885, the author Marcus Baker of the academic journal Annals of Mathematics wrote a scholarly research article titled “A Collection of Formulae for the Area of a Plane Triangle”. Encapsulated within this article are over 100 different ways to calculate the area of a triangle in a plane, and you can click here to read through the 8page article that holds this millenniaold history.
Summary
So the lecture today on everything related to calculating the area of a right triangle has come to an end. Hopefully, all of you have gained valuable and necessary knowledge in your journey to conquer the subject of Geometry. Class dismissed!
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