1. Area of a Triangle
A triangle is a polygon with three sides and three angles. The area of a triangle represents the amount of surface enclosed within its three sides.
To calculate the area, we usually need either the base and height or the three side lengths depending on the situation.
1.1. Base and Height in a Triangle
The base of a triangle can be any of its sides. The height (or altitude) is the perpendicular drawn from the opposite vertex to the base.
This perpendicular distance helps form a right angle, which is essential in calculating area using the basic formula.
2. Area Formula Using Base and Height
The most commonly used formula for the area of a triangle is:
\( A = \dfrac{1}{2} \times b \times h \)
where:
- \(b\) = base of the triangle
- \(h\) = height drawn perpendicular to the base
2.1. Working of the Formula
A triangle is exactly half of a parallelogram with the same base and height. Since the area of a parallelogram is \( b \times h \), half of it gives the triangle’s area:
\( A = \dfrac{1}{2} b h \)
2.2. Example 1
A triangle has base \(b = 12\,\text{cm}\) and height \(h = 5\,\text{cm}\). Its area is:
\( A = \dfrac{1}{2} \times 12 \times 5 = 30\,\text{cm}^2 \)
3. Area of a Triangle Using Heron's Formula
Sometimes the base or height is not given directly. In such cases, if we know the lengths of all three sides, we use Heron's formula to find the area.
3.1. Heron's Formula
If the sides of a triangle are \(a\), \(b\), and \(c\), then first compute the semi-perimeter:
\( s = \dfrac{a + b + c}{2} \)
Then the area is given by:
\( A = \sqrt{s(s - a)(s - b)(s - c)} \)
3.2. Why It Works (Simple Idea)
Heron's formula allows us to find the area of a triangle without needing height. It uses side lengths to determine how much space the triangle encloses. The product inside the square root ensures the formula only works when the three sides can form a valid triangle.
3.3. Example 2
A triangle has side lengths \(a = 7\,\text{cm}\), \(b = 8\,\text{cm}\), and \(c = 9\,\text{cm}\).
Step 1: Find the semi-perimeter:
\( s = \dfrac{7 + 8 + 9}{2} = 12 \)
Step 2: Apply Heron's formula:
\( A = \sqrt{12(12-7)(12-8)(12-9)} = \sqrt{12 \times 5 \times 4 \times 3} = \sqrt{720} = 12\sqrt{5} \)
4. Area Formula for Special Triangles
Some triangles have special properties that simplify the area calculation.
4.1. Equilateral Triangle
For an equilateral triangle of side \(a\):
\( A = \dfrac{\sqrt{3}}{4} a^2 \)
4.2. Example 3
If \(a = 6\,\text{cm}\), then:
\( A = \dfrac{\sqrt{3}}{4} \times 36 = 9\sqrt{3} \)