## How to Find the Area of a Triangle: Basic Formula and Other Methods

Learning how to find the area of a triangle is an essential skill for anyone studying mathematics or planning a career in fields such as engineering, physics, or architecture. With its simple formulas and geometric rules, finding the area of a triangle is not difficult once you understand the basic concepts. In this article, we will explore the different methods of finding the area of a triangle, including the basic formula, Heron’s formula, trigonometry, median of a triangle, triangle build method, and vector calculus. We will explain each method step-by-step, using diagrams to help you visualize the concepts. By the end of this article, you will have a better understanding of how to find the area of a triangle and be equipped with several methods to solve problems you may encounter in the future.

## Basic Formula

The basic formula for finding the area of a triangle is:

Area = 1/2 x base x height

Where the base is the bottom side of the triangle, and the height is the perpendicular distance from the base to the opposite vertex. To apply this formula, you need to measure the base and height of the triangle. For example, consider a triangle with a base of 4 cm and a height of 3 cm:

Applying the formula:

Area = 1/2 x 4 cm x 3 cm = 6 cm².

Therefore, the area of the triangle is 6 cm².

Note that the base and height can be any two sides of the triangle, not necessarily the bottom and the perpendicular side. In that case, you need to draw a perpendicular line from the vertex opposite to the base to the base to find the height.

## Heron’s Formula

Heron’s formula is an alternative method to find the area of a triangle when you don’t know the height. It works for any type of triangle, even if it is not right-angled. Heron’s formula is:

Area = √(s(s-a)(s-b)(s-c))

Where s is the semiperimeter of the triangle, i.e., half the perimeter, and a, b, and c are the lengths of the three sides. The formula derives from the Pythagorean theorem, which we will cover in the next section.

Let’s take an example of a triangle with sides 5 cm, 7 cm, and 9 cm:

Step 1: Calculate the semiperimeter s.

s = (a+b+c)/2 = (5 cm + 7 cm + 9 cm)/2 = 10.5 cm

Step 2: Apply Heron’s formula.

Area = √(10.5(10.5-5)(10.5-7)(10.5-9)) = √(10.5 x 5.5 x 3.5 x 1.5) = 11.275 cm².

Therefore, the area of the triangle is 11.275 cm².

## Trigonometry Method

If you know the length of two sides of a right-angled triangle, you can use trigonometry to find the area. Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. In a right-angled triangle, the two sides that form the right angle are called the legs, and the side opposite the right angle is called the hypotenuse. The Pythagoras theorem states that:

hypotenuse² = leg₁² + leg₂²

From this theorem, we can derive the formula to find the area of a right-angled triangle:

Area = 1/2 x leg₁ x leg₂

Let’s say you have a triangle with a base of 6 cm and a height of 4 cm:

Since the triangle is right-angled, one of the sides is the hypotenuse, and the other two are the legs. Using the Pythagoras theorem, we can find the length of the hypotenuse:

hypotenuse = √(leg₁² + leg₂²) = √(6² + 4²) = √52 = 7.211 cm

Now we can use the trigonometry formula to find the area:

Area = 1/2 x 6 cm x 4 cm = 12 cm²

Therefore, the area of the triangle is 12 cm².

## Median of a Triangle Method

Another method to find the area of a triangle that does not rely on the base and height is the median of a triangle method. A median of a triangle is a line segment that connects a vertex to the midpoint of the opposing side. A triangle has three medians, one from each vertex. The median of a triangle is useful in finding the length of the base and height of a triangle. The formula to find the area of a triangle using the median is:

Area = 1/2 x median x base

Where the median is the length of the line segment that connects a vertex to the midpoint of the opposite side, and the base is the length of the side opposite to the vertex.

Let’s take an example of a triangle with sides of 5 cm, 6 cm, and 7 cm:

Step 1: Find the midpoint of the side opposite to vertex A.

Midpoint = (B+C)/2 = (6 cm + 7 cm)/2 = 6.5 cm.

Step 2: Draw the median from vertex A to the midpoint of BC.

Step 3: Calculate the length of the median using the Pythagoras theorem.

Median = √(AB² – BM²) = √(5² – 3.25²) ≈ 3.872 cm

Step 4: Calculate the area using the formula.

Area = 1/2 x median x base = 1/2 x 3.872 cm x 5 cm = 9.68 cm².

Therefore, the area of the triangle is 9.68 cm².

## Triangle Build Method

The triangle build method is a technique to find the area of a triangle by constructing a rectangle around it. The height of the rectangle is the height of the triangle, and the base is the length of the other side. To find the length of the other side, you can use the Pythagoras theorem or any other method. Once you have the length of the base and height, you can find the area of the rectangle, which is the same as the area of the triangle.

Let’s see an example of a right-angled triangle with sides of 3 cm, 4 cm, and 5 cm:

Step 1: Draw a rectangle around the triangle, so that one side of the rectangle coincides with the base of the triangle, and the height of the triangle is the same as the height of the rectangle.

Step 2: Find the length of the other side of the rectangle, which is equal to the hypotenuse of the triangle.

Hypotenuse = √(3² + 4²) = 5 cm.

Step 3: Apply the formula for the area of a rectangle.

Area = base x height = 4 cm x 3 cm = 12 cm².

Therefore, the area of the triangle is 12 cm².

## Vector Calculation Method

The vector calculation method is a more advanced technique to find the area of a triangle. It involves using vectors and cross-products to calculate the area. Let’s first define what vectors are. A vector is a mathematical object that has a magnitude (length) and a direction. In a two-dimensional space, a vector can be represented by an arrow with a length and direction. A cross-product, on the other hand, is a mathematical operation that produces a vector that is perpendicular to two other vectors.

The formula to find the area of a triangle using vectors is:

Area = 1/2 x |a x b|

Where a and b are two vectors that represent two sides of the triangle, and |a x b| is the magnitude of the cross-product of a and b.

Let’s take an example of a triangle with vertices (3,2), (-1,1), and (2,5):

Step 1: Find the vectors a and b that represent two sides of the triangle.

a = (-1-3, 1-2) = (-4, -1)

b = (2-3, 5-2) = (-1, 3)

Step 2: Calculate the cross-product of a and b.

a x b = -4*3 – (-1)*(-1) = -11

Step 3: Find the magnitude of the cross-product.

|a x b| = √((-11)²) ≈ 11

Step 4: Apply the formula.

Area = 1/2 x |a x b| = 1/2 x 11 = 5.5

Therefore, the area of the triangle is 5.5 units².

## Conclusion

Finding the area of a triangle is an essential skill in mathematics and various fields of study, from engineering to physics to architecture. In this article, we covered several methods to find the area of a triangle, including the basic formula, Heron’s formula, trigonometry, median of a triangle, triangle build method, and vector calculus. Each method has its advantages and disadvantages, depending on the situation. Knowing these methods and being able to apply them can save time and effort when solving math problems. Practice and master these methods to become a skilled problem-solver.