Area of Polygons
Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.
First Principles
Objective: Understand and calculate the area of any regular polygon using its side length and apothem by decomposing it into congruent triangles.
Interactive Polygon Decomposer
Apothem (a): 8.66 cm
Area of 1 slice: 43.30 cm²
Total Polygon Area: 259.80 cm²
(a) Concrete Scenario: Consider a craftsman in Kisumu cutting hexagonal pavers to lay a courtyard path. To estimate the total mortar and sealant needed, he must compute the exact surface area of each tile.
(b) Geometric Principle: A regular polygon with \(n\) sides can always be partitioned from its centre into \(n\) identical isosceles triangles. For each triangle:
- The base of the triangle is the polygon's side length, \(s\).
- The perpendicular height from the centre to the midpoint of the side is the apothem, \(a\).
(c) Algebraic Formulation:
- Area of one triangle = \(\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} s a\).
- Total area of \(n\) congruent triangles = \(n \times \left(\frac{1}{2} s a\right) = \frac{1}{2} n s a\).
- Since perimeter \(P = n \times s\), the formula simplifies to \(A = \frac{1}{2} P a\).
Key Formulas
1. Area of One Triangular Sector:
\[ A_{\text{slice}} = \frac{1}{2} s a \]where \(s\) is the side length and \(a\) is the apothem (perpendicular distance from the centre to any side).
2. Total Area of a Regular Polygon (Side & Apothem):
\[ A = \frac{1}{2} n s a \]where \(n\) is the number of sides, \(s\) is the side length, and \(a\) is the apothem.
3. Total Area Using Perimeter (\(P\)):
\[ A = \frac{1}{2} P a \]where \(P = n \times s\) is the total perimeter of the regular polygon.
Worked Examples
Example 1 (Easy): Area of a Single Slice
A regular nonagon (9-sided polygon) has a side length of \(8\text{ cm}\) and an apothem of \(11\text{ cm}\). Find the area of one triangular wedge from the centre.
Step-by-step Solution:
- Identify parameters: Base \(s = 8\text{ cm}\), Height (apothem) \(a = 11\text{ cm}\).
- Apply single triangle area formula: \[ A_{\text{slice}} = \frac{1}{2} s a \]
- Substitute values: \[ A_{\text{slice}} = \frac{1}{2} \times 8 \times 11 = 4 \times 11 = 44\text{ cm}^2 \]
Answer: \(44\text{ cm}^2\)
Example 2 (Medium): Full Polygon Area from Perimeter
An artisan in Mombasa is weaving a regular hexagonal decorative mat. The total perimeter of the mat is \(48\text{ cm}\) and its apothem is \(6.93\text{ cm}\). Calculate the total area of the mat.
Step-by-step Solution:
- Identify given values: Perimeter \(P = 48\text{ cm}\), Apothem \(a = 6.93\text{ cm}\).
- Apply formula: \[ A = \frac{1}{2} P a \]
- Calculate: \[ A = \frac{1}{2} \times 48 \times 6.93 = 24 \times 6.93 = 166.32\text{ cm}^2 \]
Answer: \(166.32\text{ cm}^2\)
Example 3 (Hard): Real-World Costing and Compound Application
A luxury safari lodge in Naivasha constructs an octagonal wooden gazebo floor (8 sides). Each outer edge measures \(3\text{ m}\) and the apothem is \(3.62\text{ m}\). In the centre, an ornamental fire pit of area \(4.5\text{ m}^2\) is excluded from the wooden flooring. If high-grade teak decking costs \(1\,200\text{ KES}\) per square metre, calculate the total cost to deck the gazebo floor.
Step-by-step Solution:
- Calculate total octagonal area: \(n = 8\), \(s = 3\text{ m}\), \(a = 3.62\text{ m}\). \[ A_{\text{total}} = \frac{1}{2} n s a = \frac{1}{2} \times 8 \times 3 \times 3.62 = 12 \times 3.62 = 43.44\text{ m}^2 \]
- Subtract the fire pit area to get the decking area: \[ A_{\text{deck}} = 43.44 - 4.5 = 38.94\text{ m}^2 \]
- Calculate total cost: \[ \text{Cost} = 38.94 \times 1\,200 = 46\,728\text{ KES} \]
Answer: \(46\,728\text{ KES}\)
Common Mistakes
1. Confusing Apothem with Radius
Mistake: Using the distance from the centre to a vertex (the circumradius \(R\)) instead of the perpendicular distance to the side (the apothem \(a\)).
Why it feels right: Both line segments start at the centre, but only the apothem forms a \(90^\circ\) angle with the base, providing the true perpendicular height of each triangle slice.
Correction: Always verify that the segment meets the side at a right angle (the midpoint).
2. Forgetting the Factor of \(\frac{1}{2}\)
Mistake: Computing \(A = P \times a\) or \(n \times s \times a\), which yields exactly double the true area.
Why it feels right: It resembles the standard rectangle formula \(\text{length} \times \text{width}\).
Correction: Remember that each polygon sector is a triangle, whose area is \(\frac{1}{2} \times \text{base} \times \text{height}\).
Real World
Civil Engineering & Paving in Kenya
Interlocking concrete pavers in municipal roundabouts and plazas in Nairobi frequently use regular hexagonal geometry for maximum structural interlock. Civil engineers use \(A = \frac{1}{2} n s a\) to order exact cubic metres of concrete mix and sealants.
Apiculture (Modern Beekeeping)
Honeybee comb cells are regular hexagons that maximize storage volume while minimizing wax perimeter. Beekeepers in Kitui calculate comb cross-sectional surface area to quantify honey yield capacity per brood frame.
Architecture & Gazebo Construction
Eco-lodges across the Maasai Mara build octagonal safari tents and dining rondavels to balance 360-degree panoramic views with structural wind resistance. Estimating floor tiling and roofing thatch costs directly depends on regular polygon area formulas.
Practice