Area of Triangular and Quadrangular Land
In real-world land surveying, fields and plots are rarely perfect rectangles. They are often triangular (three-sided) or quadrangular (four-sided) in shape. Finding the area of such land requires specific geometric formulas based on what measurements are available.
A triangle is a three-sided polygon. The area of a triangular piece of land can be calculated when its base and height are known, or when all three sides are known using Heron's Formula.
A quadrangle (quadrilateral) is a four-sided polygon. For an irregular quadrilateral field, we draw a diagonal to split it into two triangles, find each triangle's area separately, and add them together. Alternatively, when a diagonal and the two perpendicular heights from opposite vertices are known, we use the direct diagonal formula.
Key Formulas:
Triangle (base & height known):
Area = ½ × base (b) × height (h)
Triangle (all three sides known — Heron's Formula):
s = (a + b + c) / 2 (semi-perimeter)
Area = √[s(s−a)(s−b)(s−c)]
Quadrangle (diagonal & two perpendicular heights):
Area = ½ × d × (h₁ + h₂)
Quadrangle (split into two triangles):
Area = Area of △₁ + Area of △₂
These problems typically involve:
- Triangular land — finding area using base and perpendicular height, or using all three sides via Heron's formula.
- Quadrangular land — splitting along a diagonal into two triangles, or using the diagonal + heights formula directly.
- Reverse problems — where the area is given and a missing dimension (base, height, diagonal, or side) must be found.
- Cost estimation — once area is found, multiplying by a rate per unit area gives total cost of ploughing, fencing, or purchasing land.
Summary of Key Relationships:
Area of Triangle = ½ × b × h
Area of Quadrangle = ½ × d × (h₁ + h₂)
Total Cost (T) = Area (A) × Rate per unit (R)
∴ A = T ÷ R and R = T ÷ A
Understanding how to calculate area of irregularly shaped land is essential in agriculture, land surveying, construction planning, and real estate — and forms a core part of mensuration in Grade 9 Mathematics.
Derivation of Area Formulas
Part 1 — Area of a Triangle (Base & Height)
A triangle with base b and perpendicular height h is exactly half of a rectangle with the same base and height.
Area of Rectangle = b × h
∴ Area of Triangle = ½ × b × h
Part 2 — Heron's Formula (All three sides known)
When the perpendicular height is not given but all three sides a, b, c are known, we use Heron's Formula:
Semi-perimeter: s = (a + b + c) / 2
Area = √[s(s − a)(s − b)(s − c)]
Example: A triangular field has sides 13 m, 14 m, and 15 m.
s = (13 + 14 + 15) / 2 = 21
Area = √[21 × (21−13) × (21−14) × (21−15)]
= √[21 × 8 × 7 × 6]
= √7056 = 84 m²
Part 3 — Area of a Quadrangle using Diagonal & Heights
Draw diagonal d across quadrangle ABCD. It splits into △ABD and △BCD. Let h₁ and h₂ be perpendicular distances from A and C to diagonal BD.
Area of △ABD = ½ × d × h₁
Area of △BCD = ½ × d × h₂
Total Area = ½ × d × h₁ + ½ × d × h₂
= ½ × d × (h₁ + h₂)
Part 4 — Cost on Land Area
Once the area of the land is calculated, the total cost of any activity (ploughing, fencing, purchasing) is:
Total Cost (T) = Area (A) × Rate per unit (R)
∴ T = A × R
A = T ÷ R (find area when cost is given)
R = T ÷ A (find rate when cost is given)
These four tools — base-height triangle formula, Heron's formula, diagonal-heights quadrangle formula, and the cost relationship — cover every problem type in this chapter.