How is the Pythagoras Theorem used to Find the Area of a Triangle?

How is the Pythagoras Theorem used to Find the Area of a Triangle?

What is the Pythagoras Theorem?

Pythagoras or Pythagorean theorem is used to give the most fundamental relationship between the sides of a right triangle. The Pythagorean theorem states that the square of the hypotenuse of a right triangle will always be equal to the sum of the square of the base and perpendicular. A note to remember is that the hypotenuse is the longest side of a right triangle. Suppose we have a triangle where the longest side measure h units and the perpendicular is p units and length, and the base is b units; then the Pythagoras formula is given by

h² = b² + p²

Let us take an example to understand how to apply the Pythagoras formula to questions.

Example: If the two sides that form the right angle in a triangle are given by 8 and 15, what is the length of the hypotenuse?

Solution: h² = b² + p²

h² = 8² + 15²

= 64 + 225

= 289

h = 289= 17 units.

Area of a Right Triangle

The area of right triangle can be described as the 2D space or the region that is enclosed within the boundaries of that triangle. The area is always represented by square units. Heron’s formula is used to find the area of a triangle whose length of all three sides is known. This formula can be manipulated to give us a simple formula to find the area of a right triangle. Suppose we have a triangle MGK. The perpendicular or height measures m units, and the base measures g units. Then the formula for finding the area of a right triangle is given by

Area = ½ * base * perpendicular

= ½ * g * m

Example: We have a triangle STW whose base measures 8 cm and perpendicular measures 10 cm. What is the area?

Solution: According to the formula

Area = ½ * base * perpendicular

= ½ * 8 * 10

= 40 cm²

However, not all questions are straightforward. Let us see an example where we have to use the Pythagoras theorem to find the area of a triangle.

Example: Find the area of a triangle with the longest side given by 17 cm, and the perpendicular is given by 8 cm.

Solution: We will break down and solve this question in two steps.

Step 1: We need to find the length of the base. For this, we need to modify and use the Pythagoras theorem.

h² = b² + p²

h² – p² = b²

b² = 17² – 8²

b² = 289 – 64 = 225

b = 225

b = 15 cm.

Step 2: As we now have the measure of the base, we can easily apply the area formula to get the answer.

Area = ½ * base * perpendicular

= ½ * 15 * 8

= 120 cm²

Applications of Pythagoras Theorem

• Pythagoras theorem is used in architecture and construction. It can be used to cut properly sized beams to support the structure.
• It is used in surveying and building maps. The theorem can be used to determine the steepness of mountains and hills.
• The theorem is also used to navigate a two-dimensional region and to find the shortest distance to the required destination.

Pythagoras Theorem: Conclusion

Right triangles and the theorems associated with them see use in all industries; hence, kids must turn to an institution such as Cuemath to build strong concepts. The math experts teach fun-filled and impactful lectures using several resources such as worksheets, math puzzles, games, apps, and visual simulations. Thus, a student is sure to master the subject under the watchful eyes of the certified math tutors.