Grade 10 essential mathematics Measurements and Geometry – Area of Polygons Notes
Essential Mathematics — 2.0 Measurements and Geometry
2.4 Area of Polygons (Age: 15, Kenyan context)
- Identify and outline the sub-sub-strands:
- Area of a triangle
- Area of a rhombus and parallelogram
- Area of a pentagon and hexagon
- Compute area of a triangle given two sides and the included acute angle (A = 1/2 ab sin C).
- Determine the area of a triangle using Heron’s formula (A = √[s(s − a)(s − b)(s − c)]).
- Determine the area of a rhombus and parallelogram in different situations (base×height, ½×product of diagonals for a rhombus, or ab sin θ).
- Determine the area of regular pentagon and hexagon from real-life situations (use apothem & perimeter or known formulas).
- Appreciate applications of polygon areas in real life (gardens, plots, roofing, tiles, fencing).
- Triangle (given two sides and included angle): A = 1/2 ab sin C
- Triangle (Heron): s = (a+b+c)/2, A = √[s(s − a)(s − b)(s − c)]
- Parallelogram: A = base × height (b × h); also A = ab sin θ
- Rhombus: A = (d1 × d2) / 2 (diagonals) or A = base × height
- Regular polygon: A = 1/2 × apothem × perimeter (A = 1/2 a P)
- Regular hexagon (side s): A = (3√3 / 2) s²
- Regular pentagon (side s): A = (1/4) √[5(5+2√5)] s² (or use apothem method)
Triangle: A = 1/2 ab sin C
Parallelogram: A = b × h
Rhombus: A = (d1 × d2) / 2
Regular hexagon: A = (3√3 / 2)s²
Regular pentagon: use A = 1/2 a P (apothem × perimeter / 2)
Example 1 — Triangle from two sides and included angle
A triangular vegetable plot has sides 8 m and 10 m with included angle 60°. Area = 1/2 × 8 × 10 × sin 60°
sin 60° = √3/2 ≈ 0.866. So A = 0.5 × 8 × 10 × 0.866 = 34.64 m² (approx).
Example 2 — Triangle by Heron’s formula
Triangle sides: 7 m, 8 m, 9 m. s = (7+8+9)/2 = 12 m.
A = √[12(12−7)(12−8)(12−9)] = √[12×5×4×3] = √[720] ≈ 26.83 m².
Example 3 — Rhombus using diagonals
Rhombus with diagonals 10 m and 8 m. Area = (10 × 8) / 2 = 40 m².
Example 4 — Regular hexagon for a tiled courtyard
If each side of a regular hexagonal tile is 0.5 m, area = (3√3 / 2) s² ≈ 2.598 × s².
A ≈ 2.598 × (0.5)² = 2.598 × 0.25 ≈ 0.6495 m² per tile.
- A triangular maize plot has sides 12 m and 15 m with included angle 45°. Find its area (give 3 s.f.).
- Find the area of a triangle with sides 13 m, 14 m, 15 m using Heron’s formula.
- A parallelogram has base 9 m and height 5 m. Find its area. If the angle between sides is 60°, check using A = ab sin θ.
- A rhombus has diagonals 14 m and 6 m. Find its area. Suggest two practical examples where this could occur.
- A regular pentagonal flower bed has side length 2 m. Use the apothem method (find apothem from geometry or approximate) to estimate area — or compute using the standard formula for regular pentagon.
- A farmer needs to fence a regular hexagonal plot with side 10 m. Find the area and the fencing length (perimeter). Give materials estimate if fencing costs KSh 250 per metre.
(Answers can be done in class; encourage calculators and neat working — units must be shown.)
- Group activity: measure a triangular bed in the school garden. Compute area using side-angle-side (use protractor) and verify using tape-measured third side and Heron’s formula.
- Practical: use cardboard to construct parallelograms and rhombuses; measure diagonals and compare area methods.
- Real-life task: estimate area of a hexagonal water tank base or paving using tiles; compute number of tiles required and cost.
- Project: design a small regular pentagon planter. Students find apothem (by geometry or trigonometry) and compute soil required (volume ≈ area × depth).
- Problem solving: use area formulas to compare cost-effectiveness of different shapes for a given enclosed area (e.g., fencing budget for hexagon vs rectangle).
- Always include units (m², cm²). Convert units before calculations if needed.
- Remember sin rule: use calculator in degree mode if angles in degrees.
- For Heron’s formula, compute s first and check that (s − a), (s − b), (s − c) are positive.
- For regular polygons, apothem is perpendicular distance from centre to a side; A = 1/2 a P is often simpler than a messy s-only formula.
- When given diagonals for rhombus, use A = d1 × d2 / 2. Do not confuse with rectangle formula.
- Agriculture: calculating planting area for triangular or irregular plots on a farm in Kisumu or Nakuru.
- Construction: estimating cement or tiles for pentagonal or hexagonal features on a house in Nairobi.
- Urban planning: allocating playground space with polygonal shapes.
- Fencing and costs: comparing fencing lengths and areas to choose the most economical shape for a given land size.
Knowing several methods to find areas (side-angle-side, Heron, base×height, diagonals, apothem×perimeter) helps solve many practical problems. Always choose the formula that fits the given data, check units, and interpret results in context (cost, material, planting, etc.).