How Do You Derive the Area of a Triangle?

The area of a triangle is derived geometrically, most commonly by relating it to the area of a rectangle or parallelogram. The fundamental formula for the area of any triangle is half of the base times the height.

Understanding the Basic Formula

Based on the provided reference, the area (A) of a triangle is calculated using the formula:

A = 1/2 × b × h

Where:

• b represents the length of the base of the triangle.
• h represents the height of the triangle, which is the perpendicular distance from the base to the opposite vertex.

To find the area of a triangle using this formula, you essentially need to know these two key measurements: its base and its corresponding height.

How the Formula is Derived

The formula A = 1/2 × b × h can be visualized and derived from the area of a rectangle or parallelogram.

1. From a Rectangle:

   • Consider a rectangle with base b and height h. Its area is simply b × h.
   • If you draw a diagonal line across the rectangle, you divide it into two congruent right-angled triangles.
   • Each of these right-angled triangles has a base b and a height h.
   • Since the two triangles make up the entire rectangle, the area of one triangle must be exactly half the area of the rectangle.
   • Therefore, the area of a right-angled triangle is (b × h) / 2, or 1/2 × b × h.

2. From a Parallelogram:

   • Consider a parallelogram with base b and height h. Its area is b × h.
   • If you draw a diagonal across the parallelogram, you divide it into two congruent triangles.
   • Each of these triangles has a base b and a height h (the perpendicular distance between the base and the opposite side).
   • Since the two triangles make up the entire parallelogram, the area of one triangle must be half the area of the parallelogram.
   • Therefore, the area of a triangle derived from a parallelogram is also (b × h) / 2, or 1/2 × b × h.

This derivation shows that any triangle can be seen as half of a parallelogram (or rectangle, in the case of right triangles) with the same base and height.

Applicability to All Triangle Types

As stated in the reference, the formula A = 1/2 × b × h is universally applicable:

• Scalene Triangles: Triangles with all sides of different lengths.
• Isosceles Triangles: Triangles with two sides of equal length.
• Equilateral Triangles: Triangles with all sides of equal length.
• Right-Angled Triangles: Triangles with one 90-degree angle (where one leg can serve as the base and the other as the height).
• Acute Triangles: Triangles where all angles are less than 90 degrees.
• Obtuse Triangles: Triangles with one angle greater than 90 degrees (the height may fall outside the triangle itself when using a specific base).

The key is always to use the perpendicular height corresponding to the chosen base.

Calculating Area: Step-by-Step

To calculate the area of a triangle using the base and height:

1. Identify the Base (b): Choose one side of the triangle to be the base.
2. Identify the Height (h): Find the perpendicular distance from the opposite vertex to the chosen base (or the extension of the base for obtuse triangles).
3. Apply the Formula: Substitute the values of b and h into the formula A = 1/2 × b × h and calculate.

Example:

Consider a triangle with a base of 10 cm and a height of 6 cm.

• b = 10 cm
• h = 6 cm

Area (A) = 1/2 × 10 cm × 6 cm
Area (A) = 1/2 × 60 cm²
Area (A) = 30 cm²

The area of the triangle is 30 square centimeters.