Area of triangles and polygons Knowledge-Based Study

Study notes

1. Understanding the Area of a Triangle

   The area of any triangle measures the two-dimensional space enclosed by its three sides. The fundamental formula for calculating the area of a triangle is derived directly from the area of a parallelogram.

   For any triangle with base b and perpendicular height h corresponding to that base, the area A is given by:

   $A = \frac{1}{2} b h$

   Key Points to Remember

   • Base (b): Any side of the triangle can be chosen as the base.
   • Height (h): The perpendicular distance from the opposite vertex to the line containing the chosen base.
   • In a right-angled triangle, the two legs are mutually perpendicular, so one leg serves as the base and the other as the height: $A = \frac{1}{2} a b$.
   • In an obtuse-angled triangle, the altitude to a side adjacent to the obtuse angle falls outside the triangle onto the extension of the base line.

   Example: If a triangle has a base of 14 cm and an altitude corresponding to that base measuring 9 cm, its area is:

   $A = \frac{1}{2} \times 14 \times 9 = 63\text{ cm}^2$

   Parallelograms, Trapezoids, and Rhombuses

   Special quadrilaterals have standard area formulas determined by their geometric properties.

   1. Parallelogram

   A parallelogram can be rearranged into a rectangle of equal base and height.

   $A = b h$

   where b is the length of the base and h is the perpendicular height between the parallel bases.

   2. Trapezoid (Trapezium)

   A trapezoid has one pair of parallel sides, often called bases a and b. Its area is equal to the average of the two bases multiplied by the perpendicular height h:

   $A = \frac{a + b}{2} h$

   3. Rhombus and Kite

   For any quadrilateral whose diagonals are perpendicular, such as a rhombus or a kite, the area is half the product of the lengths of its diagonals $d_1$ and $d_2$:

   $A = \frac{1}{2} d_1 d_2$

   Example: A trapezoid has parallel bases of lengths 8 m and 12 m, and a perpendicular distance between them of 5 m. Its area is:

   $A = \frac{8 + 12}{2} \times 5 = 10 \times 5 = 50\text{ m}^2$

   Regular Polygons and the Apothem

   A regular polygon is both equilateral (all sides are equal in length) and equiangular (all interior angles are equal). Common regular polygons include equilateral triangles, squares, regular pentagons, and regular hexagons.

   The Apothem

   The apothem (denoted as a) is the perpendicular segment drawn from the center of a regular polygon to the midpoint of any of its sides.

   Formula for Area

   A regular polygon with n sides of length s can be partitioned from its center into n congruent isosceles triangles. The base of each triangle is s and the height is the apothem a.

   The area of each small triangle is $\frac{1}{2} s a$. Since there are n such triangles, the total area is:

   $A = n \times \left(\frac{1}{2} s a\right) = \frac{1}{2} a (n s)$

   Because the perimeter P equals $n s$, we obtain the standard formula:

   $A = \frac{1}{2} a P$

   Example: A regular hexagon has a side length of 6 cm and an apothem of approximately 5.2 cm. Its perimeter is $P = 6 \times 6 = 36\text{ cm}$. Its area is:

   $A = \frac{1}{2} \times 5.2 \times 36 = 93.6\text{ cm}^2$

   Finding the Area of Composite Polygons

   Many geometric figures are composite shapes formed by joining or removing standard polygons. Two primary strategies are used to compute their areas:

   1. Additive Method (Decomposition)

   Split the irregular polygon into non-overlapping standard shapes whose areas can be calculated individually, then sum their areas:

   $A_{\text{total}} = A_1 + A_2 + \dots + A_k$

   • Identify natural boundary lines such as vertices and right angles.
   • Subdivide the figure into triangles, rectangles, or trapezoids.
   • Calculate each sub-area and add them together.

   2. Subtractive Method

   Enclose the irregular polygon within a larger standard shape (such as a rectangle), then subtract the areas of the unshaded or outer regions:

   $A_{\text{total}} = A_{\text{bounding box}} - A_{\text{unwanted regions}}$

   This method is especially effective on coordinate grids or when the surrounding shape is a simple rectangle.