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Learning Resources

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.

Grade 10 Pathway: N/A

First Principles

Objective: Calculate the area of regular polygons using decomposition into triangles.

Concrete scenario: A hexagonal gazebo floor in a community garden in Nakuru needs tiling. To know how many tiles to buy, you must find the floor area. Instead of measuring the whole hexagon at once, imagine cutting it into triangular slices from the centre — like cutting a round flatbread into equal slices. Each slice is an isosceles triangle, and you already know how to find the area of a triangle.

Geometric insight: Every regular polygon can be split into \(n\) identical isosceles triangles, where \(n\) is the number of sides. Each triangle has its base along one side of the polygon and its apex at the centre. The perpendicular distance from the centre to a side is called the apothem (\(a\)) — it acts as the exact height of every triangular slice.

Algebraic rule: If each triangle has base \(s\) (side length) and height \(a\) (apothem), then one triangle's area is \(\frac{1}{2} s a\). With \(n\) identical triangles, the total area is: \[A = n \times \left(\frac{1}{2} s a\right) = \frac{1}{2} (n s) a = \frac{1}{2} P a\] where \(P = n \times s\) is the perimeter of the regular polygon.

Side length (\(s\)): —
Perimeter (\(P\)): —
One slice area: —
Total polygon area: —

Key Formulas

\[A = \frac{1}{2} \times P \times a\] — The area of any regular polygon equals half the perimeter multiplied by the apothem. Here, Perimeter \(P = n \times s\), where \(n\) is the number of sides and \(s\) is the side length.
\[A_{\text{triangle slice}} = \frac{1}{2} \times s \times a\] — Each of the \(n\) identical triangular slices created from the centre has base \(s\) and height \(a\) (apothem).
\[A = n \times A_{\text{slice}} = n \times \left(\frac{1}{2} \times s \times a\right) = \frac{1}{2} n s a\] — Summing the areas of all \(n\) congruent triangular slices gives the complete polygon area.

Worked Examples

Example 1 (Easy): Regular Pentagon
A regular pentagon has a side length \(s = 6\text{ cm}\) and an apothem \(a = 4.13\text{ cm}\). Find its total area.
  1. Identify given values: Number of sides \(n = 5\), side length \(s = 6\text{ cm}\), apothem \(a = 4.13\text{ cm}\).
  2. Calculate perimeter (\(P\)): \[P = n \times s = 5 \times 6 = 30\text{ cm}\]
  3. Apply polygon area formula: \[A = \frac{1}{2} \times P \times a = \frac{1}{2} \times 30 \times 4.13\]
  4. Compute result: \[A = 15 \times 4.13 = 61.95\text{ cm}^2\]
Example 2 (Medium): Hexagonal Paver using Decomposition
A decorative regular hexagonal paving stone has a side length \(s = 8\text{ cm}\) and an apothem \(a = 6.93\text{ cm}\). Find its area by calculating the area of one triangular slice first.
  1. Identify slice dimensions: The base of one triangular slice is \(s = 8\text{ cm}\) and the perpendicular height is \(a = 6.93\text{ cm}\).
  2. Calculate area of one slice: \[A_{\text{slice}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 6.93 = 27.72\text{ cm}^2\]
  3. Multiply by the total number of slices: A regular hexagon splits into \(n = 6\) equal triangles. \[A_{\text{total}} = 6 \times A_{\text{slice}} = 6 \times 27.72 = 166.32\text{ cm}^2\]
Example 3 (Hard): Octagonal Gazebo Floor & Tiling
A school gazebo in Nairobi is built as a regular octagon with side length \(s = 2.5\text{ m}\) and apothem \(a = 3.04\text{ m}\). Find the floor area, and determine how many square tiles measuring \(0.5\text{ m} \times 0.5\text{ m}\) (each \(0.25\text{ m}^2\)) are required to cover the entire floor.
  1. Determine perimeter: \[P = 8 \times 2.5 = 20\text{ m}\]
  2. Compute total floor area: \[A = \frac{1}{2} \times P \times a = \frac{1}{2} \times 20 \times 3.04 = 10 \times 3.04 = 30.4\text{ m}^2\]
  3. Determine number of tiles: \[\text{Tiles required} = \frac{\text{Total Area}}{\text{Tile Area}} = \frac{30.4}{0.25} = 121.6\]
  4. Practical rounding: Since partial tiles cannot be purchased, round up to \(122\) tiles.

Common Mistakes

Mistake Confusing the side length \(s\) with the apothem \(a\), or writing \(A = \frac{1}{2} P s\).
Correction The apothem is the perpendicular distance from the centre of the polygon to the midpoint of any side. It represents the height of each interior triangle, while \(s\) is the base along the polygon edge. Always use \(A = \frac{1}{2} P a\).
Why it feels right The side length is the most noticeable exterior measurement, so students often mistakenly substitute it for the apothem.
Mistake Forgetting the factor of \(\frac{1}{2}\) and calculating area as \(P \times a\) or \(s \times a\).
Correction Regular polygon decomposition forms triangles, not rectangles. Each triangle's area is \(\frac{1}{2} s a\), so the total area must include the factor of \(\frac{1}{2}\).
Why it feels right Rectangular area formulas (\(\text{length} \times \text{width}\)) are heavily memorised, leading learners to omit the \(\frac{1}{2}\) required for triangular decomposition.

Real World

Gazebos and Bandstands: In urban parks across Nairobi and Kisumu, octagonal gazebos are built using regular polygonal concrete bases. Decomposing the floor into 8 triangles allows engineers to precisely calculate floor screed and tile requirements.
Beehive Honeycomb Architecture: Honeycomb cells are regular hexagons. Calculating the cross-sectional area using the apothem helps apiculturists and designers model maximum storage efficiency with minimal wax perimeter.
Urban Paver Production: Precast concrete manufacturing companies produce interlocking hexagonal pavers. Determining the exact surface area ensures accurate costing for cement, pigment, and sand per block.
Nut & Bolt Engineering: Mechanical engineers use regular hexagonal bolts and nuts. Calculating the top surface area using side length and apothem determines the bearing surface area available to distribute clamping force.

Practice

A regular pentagon has a perimeter of 40 cm and an apothem of 5.5 cm. Calculate its area in cm². (Type only the number, e.g., 42)
Review the concepts above.
A regular hexagon has an apothem of 6 cm and each side length is 7 cm. What is the area of just one of its 6 triangular slices in cm²? (Type only the number, e.g., 42)
Review the concepts above.
A community garden in Nakuru is designed as a regular octagon (8 sides). Each side measures 5 m and the apothem is 6 m. What is the total area of the garden in m²? (Type only the number, e.g., 42)
Review the concepts above.
A decorative badge shaped as a regular nonagon (9 sides) has a total perimeter of 54 cm and an apothem of 8.2 cm. Calculate the total area of the badge in cm². (Type only the number, e.g., 42.5)
Review the concepts above.
A hotel in Mombasa constructs an outdoor pavilion with a regular hexagonal floor. Each side of the hexagon is 6 metres and the apothem is 5.2 metres. The stone tiles cost 1500 KES per square metre. What will be the total cost of the tiles in KES? (Type only the number, e.g., 42000)
Review the concepts above.
A cultural centre has a regular decagonal (10-sided) wooden platform. Each side measures 4 m and the apothem is 6.15 m. If 1 litre of wood sealant covers 5 m² of area, how many full 1-litre cans of sealant must be purchased to provide 1 complete coat on the entire platform? (Round up to the nearest whole can.) (Type only the number, e.g., 42)
Review the concepts above.