Triangles are one of the most fundamental shapes in geometry. Their properties have fascinated mathematicians for centuries, and they appear everywhere in both pure mathematics and practical applications, from architecture to engineering. One common problem involving triangles is figuring out the area when you know two sides and the angle between them. This blog post will take an in-depth look at how to calculate the area of a triangle in such situations, why the formula works, and some practical examples.
Triangle Area Calculator (2 Sides & 1 Angle)
Introduction to Triangles and Area Calculation
A triangle is a three-sided polygon with three angles. The sum of its interior angles is always one hundred eighty degrees. The area of a triangle is simply the amount of space enclosed within its three sides. Calculating the area of a triangle can be done in multiple ways depending on what information is available. For example, if the base and height are known, the area is half the product of the base and the height.
However, sometimes the height may not be directly known, but other parts of the triangle, such as two sides and the angle between them, are given. This is a common case in trigonometry and can be especially useful when dealing with oblique triangles, where the height is not given or is difficult to find.
The Two Sides and Included Angle Situation
Imagine a triangle where two sides are known, say side A and side B. Additionally, the angle between these two sides is also known, which we will call angle C. Importantly, the angle is the included angle, meaning it is the angle formed where the two known sides meet.
This trio of pieces of data — two sides and the included angle — is enough to determine the area of the triangle using a specific formula derived from trigonometry, without needing the height directly.
The Formula for Area With Two Sides and the Included Angle
The formula to calculate the area of a triangle, knowing two sides and the included angle, is:
Area equals one half times the product of the two sides times the sine of the included angle.
Breaking this down, the area is half of the multiplication of side A and side B, multiplied by the sine of angle C.
Why Does This Formula Work?
To understand why this formula works, let’s revisit the basic formula for area — half the base times the height.
In a triangle with sides A and B and the included angle C, one side can be treated as the base. Then the height is the perpendicular distance from the opposite vertex to the base.
However, if we do not have the height, we can express the height in terms of the other side and the sine of the angle.
Consider side B as the base. Drop a perpendicular from the vertex opposite this side to the base. This perpendicular is the height, h.
Using trigonometry, h equals A times the sine of C, because the sine of angle C is the ratio of the height over side A.
Substituting the height h into the area formula:
Area equals one half times the base times the height equals one half times B times A times sine of C.
This is the exact formula we use.
Step-by-Step Guide to Using the Formula
Here’s how you can apply this formula in practice:
- Identify the two known sides of the triangle.
- Identify the included angle between these two sides.
- Find the sine of the included angle. This can be done using a scientific calculator or sine tables.
- Multiply the two known sides.
- Multiply the result by the sine of the included angle.
- Multiply by one half.
- The answer is the area of the triangle.
Important Points to Remember
- The angle must be the included angle between the two known sides for the formula to work directly.
- The sine of the angle should be expressed in degrees or radians based on your calculator’s mode, but consistency is key.
- This formula is very useful when the height is not readily measurable, which is common in irregular or oblique triangles.
Practical Example
Let’s put this into practice with an example.
Suppose you are given a triangle where one side measures ten meters, another side measures fifteen meters, and the angle between them is thirty degrees.
Step one: two sides are ten meters and fifteen meters.
Step two: the angle between is thirty degrees.
Step three: find the sine of thirty degrees. The sine of thirty degrees is known to be point five or 1/2.
Step four: multiply the two sides: ten times fifteen equals one hundred fifty.
Step five: multiply the result by the sine of the angle: one hundred fifty times 0.5 equals seventy-five.
Step six: take half of this: half of seventy-five is thirty-seven point five.
Thus, the area of the triangle is thirty-seven point five square meters.
More Examples for Practice
Example Two: Sides are eight feet and twelve feet with an included angle of forty-five degrees.
Sine of forty-five degrees is approximately 0.707.
Calculate area: one half times eight times twelve times 0.707 = one half times ninety-six times 0.707 = forty-eight times 0.707 ≈ thirty-four square feet.
Example Three: Sides fifteen centimeters and twenty centimeters with an included angle of sixty degrees.
Sine of sixty degrees is about 0.866.
Area: one half times fifteen times twenty times 0.866 = one half times three hundred times 0.866 = one hundred fifty times 0.866 = one hundred twenty-nine point nine square centimeters.
Real-Life Applications
This formula is very practical in many real-world problems. For instance, in construction and architecture, when designing roof slopes or triangular trusses, the exact area might be needed but only side lengths and one angle are often available.
In navigation and surveying, when determining land plots or irregular areas, knowing two distances and an angle can help calculate areas without having to measure height directly.
Alternative Methods and When They Are Useful
If you know all three sides, you might use Heron’s formula, which involves the semi-perimeter and all three sides.
If the height is available, the classic base times height divided by two formula is simplest.
When knowing two sides but not the included angle, or when the given angle is not between the two known sides, this formula cannot be directly applied.
Summary
To sum up, the area calculation of a triangle when two sides and the included angle are known is a straightforward method based on trigonometry. The formula is one half times the product of the two sides times the sine of the included angle.
This approach helps find areas in non-right angled triangles where height is difficult to ascertain directly but side lengths and angles are known. By understanding and applying this formula, students, engineers, architects, and surveyors can solve numerous practical problems efficiently.
Final Thoughts
Triangles are versatile and studying their properties enriches our understanding of geometry and its applications. Knowing how to calculate area with just two sides and an included angle adds an important tool to your math toolkit, making you better prepared to tackle complex problems in math, science, and engineering. Practice with different side lengths and angles will improve your fluency and confidence with this formula.