Finding the Side Length of an Equilateral Triangle with an Area of 1.5 cm2
Haven't tackled one of these in a while! Upon revisiting this problem, we often find ourselves needing to determine the side length of an equilateral triangle given its area. In this context, we are dealing with a 1.5 cm2 equilateral triangle, and in this article, we'll explore how to find the side length step-by-step.
Understanding the Basics
Equilateral triangles are a special type of triangle where all three sides and all three internal angles are equal. Each angle in an equilateral triangle measures 60°.
The area of any triangle can be calculated using the formula:
Area 1/2 x base x height
However, given the complexity of an equilateral triangle, direct application of the base-height formula can be tricky. Instead, trigonometric relationships come in handy.
Using Trigonometry
In an equilateral triangle, the altitude (height) divides the base into two equal halves. If we let the side length of the equilateral triangle be s, then the base is s, and the height is the altitude, which can be expressed using trigonometry:
Height (h) s * sin(60°) s * √3 / 2
Substituting the height into the area formula, we have:
1.5 1/2 * s * (s * √3 / 2)
Let's simplify this equation step-by-step:
Step-by-Step Solution
Simplify the equation inside the parentheses:
1.5 1/2 * s2 * √3 / 2
Multiply the numerator on the right-hand side:
1.5 (s2 * √3) / 4
Clear the fraction by multiplying both sides by 4:
6 s2 * √3
Solve for s2 by dividing both sides by √3:
s2 6 / √3
Simplify the right-hand side:
s2 2 * √3
Take the square root of both sides to solve for s:
s √(2 * √3)
Equivalent to:
s ≈ 1.633 cm
Conclusion
Through the use of trigonometric relationships and algebraic manipulation, we have determined that the side length of an equilateral triangle with an area of 1.5 cm2 is approximately 1.633 cm. This process highlights the usefulness of understanding geometric principles and trigonometry in solving complex problems.
Related Keywords
equilateral triangle area side lengthExplore Further
If you're interested in more geometric concepts and problem-solving techniques, consider exploring topics such as:
The area and properties of equilateral triangles Using trigonometry to solve geometric problems Basic algebra and its applications in geometryBy delving into these areas, you can enhance your mathematical skills and deepen your understanding of geometric principles.