Understanding the Area of an Obtuse-Angled Triangle

An obtuse-angled triangle is a triangle with one angle greater than 90 degrees but less than 180 degrees. This article will explore the methods and formulas for calculating the area of such triangles.

General Area Formula for an Obtuse Triangle

The area of an obtuse-angled triangle, like any other type of triangle, can be calculated using the same formula: Area 1/2 x base x height. This formula is universal and can be applied regardless of which side is chosen as the base, as long as the corresponding height is accurately determined.

Methods to Determine the Area

There are three possible ways to calculate the area of an obtuse triangle, each corresponding to a different side of the triangle being considered as the base. Let's consider a triangle ABC with sides AB, AC, and BC, where AB is chosen as the base. The height, which is the perpendicular from C to AB, will determine the area. Specifically, the areas can be calculated as follows:

Area 1/2 x AB x CE Area 1/2 x AC x AH Area 1/2 x AC x BC

In practice, the chosen base and its corresponding height will depend on the triangle's specific dimensions and location of the vertex angles.

Using Heron's Formula for Area Calculation

For certain obtuse triangles, particularly those with sides of varying lengths, Heron's formula provides an alternative method for determining the area. Heron's formula is given as:

A √[ss - (s - a)(s - b)(s - c)]

Where:

a, b, and c are the lengths of the sides of the triangle. s is the semi-perimeter of the triangle, calculated as: s (a b c) / 2.

Using this formula, you can input the side lengths to find the area of the obtuse triangle. For example, for a triangle with sides 9, 10, and 17, the area can be calculated easily:

A √[s(s - 9)(s - 10)(s - 17)]
Where s (9 10 17) / 2 18,
Resulting in A √[18(9)(8)(1)] 36.

Special Cases Where the Perpendicular Falls Outside the Triangle

In some obtuse triangles, the height calculation becomes more complex. If the base is the longest side, the height may not fall within the triangle, leading to a need for extending the base to meet the perpendicular. For instance, in an obtuse triangle where the base is the longest side, extending the base to find the height will allow for the area calculation using the formula: Area 1/2 x base x height.

Example with Given Values

Consider a triangle ABC where AB 8 cm, AC 12 cm, and angle A 150°. We know that sin 150° 1/2. Therefore, the area of the triangle can be calculated as:

Area 1/2 x b x c x sin A 1/2 x 8 x 12 x 1/2 24 cm2.

This example demonstrates how the standard area formula with the sine function can be applied in obtuse triangles.