How to find the area of a triangle given 3 sides?
Understand the Problem
The question is asking for a method to calculate the area of a triangle when the lengths of all three sides are known. This involves using Heron's formula, which first calculates the semi-perimeter and then uses it to find the area.
Answer
The area of the triangle is given by $A = \sqrt{s(s - a)(s - b)(s - c)}$ where $s = \frac{a + b + c}{2}$.
Answer for screen readers
The area of the triangle can be expressed as:
$$ A = \sqrt{s(s - a)(s - b)(s - c)} $$
Where (s = \frac{a + b + c}{2}).
Steps to Solve
- Calculate the Semi-perimeter
To find the semi-perimeter (s) of the triangle, we use the formula:
$$ s = \frac{a + b + c}{2} $$
where (a), (b), and (c) are the lengths of the three sides of the triangle.
- Apply Heron's Formula for Area
Once we have the semi-perimeter, we can calculate the area (A) using Heron's formula:
$$ A = \sqrt{s(s - a)(s - b)(s - c)} $$
This formula requires substituting (s), (a), (b), and (c) from the previous steps.
- Calculate the Area
Once all variables are substituted in, compute the square root to find the area of the triangle.
The area of the triangle can be expressed as:
$$ A = \sqrt{s(s - a)(s - b)(s - c)} $$
Where (s = \frac{a + b + c}{2}).
More Information
Heron's formula is a powerful tool for finding the area of a triangle when you know the lengths of all three sides, which is particularly useful when the triangle's height is not easily measured.
Tips
- Miscalculating the semi-perimeter. Ensure you divide the sum of the sides by 2 accurately.
- Forgetting to apply the square root correctly. Make sure to compute (A) using the square root of the product properly.
- Not checking if the triangle inequality holds, which ensures that a triangle can actually be formed with the given side lengths.
