What is the formula for the area of a triangle?
Understand the Problem
The question is asking for the formula to calculate the area of a triangle. The formula depends on what information you have about the triangle (e.g., base and height, three sides, two sides and an included angle).
Answer
The area of a triangle can be calculated as: $A = \frac{1}{2}bh$, $A = \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \frac{a+b+c}{2}$, or $A = \frac{1}{2}ab\sin(C)$.
Answer for screen readers
The formulas for the area of a triangle are:
- Using base and height: $A = \frac{1}{2}bh$
- Using three sides (Heron's formula): $A = \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{a + b + c}{2}$
- Using two sides and an included angle: $A = \frac{1}{2}ab\sin(C)$
Steps to Solve
- Area using base and height
If you know the base ($b$) and height ($h$) of the triangle, the area ($A$) is half the product of the base and height.
$$ A = \frac{1}{2}bh $$
- Area using three sides (Heron's formula)
If you know the lengths of the three sides ($a$, $b$, $c$), you can use Heron's formula. First, calculate the semi-perimeter ($s$).
$$ s = \frac{a + b + c}{2} $$
Then, the area ($A$) is given by:
$$ A = \sqrt{s(s-a)(s-b)(s-c)} $$
- Area using two sides and an included angle
If you know the length of two sides (e.g., $a$ and $b$) and the included angle ($C$) between them, the area ($A$) is:
$$ A = \frac{1}{2}ab\sin(C) $$
The formulas for the area of a triangle are:
- Using base and height: $A = \frac{1}{2}bh$
- Using three sides (Heron's formula): $A = \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{a + b + c}{2}$
- Using two sides and an included angle: $A = \frac{1}{2}ab\sin(C)$
More Information
The area of a triangle can be found using different formulas depending on the available information. Heron's formula is especially useful when only the lengths of the three sides are known. The formula using two sides and an included angle is derived from trigonometry.
Tips
A common mistake is using the wrong formula for the given information. For example, trying to use the base-height formula when you only know the three sides. Another mistake is incorrectly calculating the semi-perimeter in Heron's formula or forgetting to take the square root at the end. Also, make sure the angle is in degrees or radians depending on your calculator settings when using the formula involving sine.
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