How to find the height of a triangle with 3 sides?
Understand the Problem
The question is asking for a method to calculate the height of a triangle when the lengths of all three sides are known. This likely involves using Heron's formula to first find the area of the triangle and then using the relationship between the area, base, and height to solve for the height.
Answer
$$ h = \frac{2 \sqrt{s(s - a)(s - b)(s - c)}}{b} $$ where $s = \frac{a + b + c}{2}$.
Answer for screen readers
The height (h) of the triangle can be calculated with the formula:
$$ h = \frac{2 \sqrt{s(s - a)(s - b)(s - c)}}{b} $$
where (s = \frac{a + b + c}{2}).
Steps to Solve
- Identify the sides of the triangle
Let's denote the lengths of the sides of the triangle as $a$, $b$, and $c$.
- Calculate the semi-perimeter
The semi-perimeter (s) is calculated using the formula:
$$ s = \frac{a + b + c}{2} $$
This value will be used in Heron's formula.
- Apply Heron's formula to find the area
Now we can calculate the area (A) of the triangle using Heron's formula:
$$ A = \sqrt{s(s - a)(s - b)(s - c)} $$
This calculates the area based on the semi-perimeter and the lengths of the sides.
- Choose a base and calculate the height
Choose one of the sides as the base (let's say (b)). To find the height (h) corresponding to this base, we can use the formula relating area, base, and height:
$$ A = \frac{1}{2} \times b \times h $$
Rearranging to solve for (h):
$$ h = \frac{2A}{b} $$
- Plug the area into the height formula
Substituting for (A) from Heron's formula into the height equation:
$$ h = \frac{2 \sqrt{s(s - a)(s - b)(s - c)}}{b} $$
This will give you the height of the triangle corresponding to the chosen base.
The height (h) of the triangle can be calculated with the formula:
$$ h = \frac{2 \sqrt{s(s - a)(s - b)(s - c)}}{b} $$
where (s = \frac{a + b + c}{2}).
More Information
Heron's formula is a powerful tool in geometry that allows for the calculation of the area of a triangle when only the lengths of its sides are known. Once you know the area, you can find the height relative to any side used as a base.
Tips
- Not using the correct semi-perimeter formula.
- Forgetting to take the square root in Heron's formula.
- Confusing the base of the triangle for height calculations.
