1. What is the Area of a Triangle?
The area of a triangle tells us how much space it covers on a flat surface. It is measured in square units (like cm², m², etc.).
Different types of triangles or different sets of given information may require different formulas to find the area.
2. Formula 1: Using Base and Height
This is the most commonly used formula and applies to all types of triangles.
Area formula:
\( \text{Area} = \dfrac{1}{2} \times \text{base} \times \text{height} \)
2.1. Understanding Base and Height
The base can be any side of the triangle. The height (also called altitude) is the perpendicular drawn from the opposite vertex to that base.
A
/|
/ | h (height)
/ |
B---C ← base
2.2. Example
If the base is \(10\,\text{cm}\) and height is \(6\,\text{cm}\):
\( \text{Area} = \dfrac{1}{2} \times 10 \times 6 = 30\,\text{cm}^2 \)
3. Formula 2: Heron’s Formula (Using Three Sides)
Heron’s formula is useful when all three sides of the triangle are known, but the height is not given.
First compute the semi-perimeter:
\( s = \dfrac{a + b + c}{2} \)
Then:
\( \text{Area} = \sqrt{s(s - a)(s - b)(s - c)} \)
3.1. Example
If the sides are \(a = 7\), \(b = 8\), \(c = 9\):
\( s = \dfrac{7 + 8 + 9}{2} = 12 \)
\( \text{Area} = \sqrt{12(12-7)(12-8)(12-9)} = \sqrt{12 \cdot 5 \cdot 4 \cdot 3} = 12\sqrt{5} \)
4. Formula 3: Area of a Right Triangle
For a right-angled triangle, the two legs (the sides forming the right angle) act as the base and height.
If \(a\) and \(b\) are the legs, then:
\( \text{Area} = \dfrac{1}{2}ab \)
|
a |\ b
| \
----
c (hyp.)
5. Formula 4: Using Two Sides and Included Angle
If two sides and the angle between them are known, the area can be found using:
\( \text{Area} = \dfrac{1}{2}ab\sin C \)
Useful in non-right triangles.
6. Formula 5: Area using Coordinates
If the triangle’s vertices are given as coordinates:
\(A(x_1, y_1),\ B(x_2, y_2),\ C(x_3, y_3)\)
The area formula is:
\( \text{Area} = \dfrac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \)
6.1. Why This Works
This formula comes from the determinant method, which calculates the area of a polygon using coordinate geometry. It works for any triangle drawn on the x-y plane.
7. Choosing the Right Formula
Use the formula that matches the information given:
- Base + height → \( \dfrac{1}{2}bh \)
- Three sides → Heron’s formula
- Right triangle → \( \dfrac{1}{2}ab \)
- Two sides + included angle → \( \dfrac{1}{2}ab\sin C \)
- Coordinates → determinant formula
8. Applications of Triangle Area
Area formulas are used in:
- Construction and architecture
- Surveying land
- Mapping and navigation
- Physics and engineering
- Designing triangular components and patterns