Area of Triangular and Quadrilateral Shaped Land
In land measurement, fields and plots are rarely perfect rectangles. They are often triangular or quadrilateral (four-sided) in shape. To find the area of such land, we use specific geometric formulas depending on the shape and the measurements available.
A triangle is a three-sided polygon. The area of a triangular piece of land can be found if its base and height are known, or if all three sides are known (using Heron's formula).
A quadrilateral is a four-sided polygon. It includes shapes like rectangles, parallelograms, trapeziums, and irregular quadrilaterals. For an irregular quadrilateral field, we often split it into two triangles using a diagonal, find each triangle's area separately, and add them together.
Key Formulas:
Triangle (base & height known): Area = ½ × base (b) × height (h)
Triangle (all sides known — Heron's Formula):
s = (a + b + c) / 2 (semi-perimeter)
Area = √[s(s−a)(s−b)(s−c)]
Quadrilateral (diagonal & two heights): Area = ½ × d × (h₁ + h₂)
Quadrilateral (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 all three sides via Heron's formula.
- Quadrilateral land — splitting along a diagonal into two triangles, then summing their areas.
- Reverse problems — where the area is given and a missing dimension (base, height, or side) must be found.
- Cost estimation on land — once the area is found, multiplying by a rate per unit area to find total cost of fencing, cultivating, or purchasing.
Summary of Key Relationships:
Area of Triangle = ½ × b × h
Area of Quadrilateral = ½ × d × (h₁ + h₂)
Total Cost (T) = Area (A) × Rate (R)
∴ A = T ÷ R and R = T ÷ A
Understanding the area of irregularly shaped land is essential in agriculture, surveying, construction, and real estate — and forms a core part of mensuration in Grade 9 Mathematics.
Derivation of Area Formulas
Part 1 — Area of a Triangle
A triangle with base b and perpendicular height h can be seen as 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 (when all three sides are known)
When the 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 triangle has sides 5m, 6m, and 7m.
s = (5 + 6 + 7) / 2 = 9
Area = √[9 × (9−5) × (9−6) × (9−7)]
= √[9 × 4 × 3 × 2]
= √216 ≈ 14.70 m²
Part 3 — Area of a Quadrilateral using a Diagonal
Draw diagonal d across the quadrilateral ABCD. This splits it into two triangles. Let h₁ and h₂ be the perpendicular heights from the opposite vertices to the diagonal.
Area of △₁ = ½ × d × h₁
Area of △₂ = ½ × d × h₂
Total Area = ½ × d × h₁ + ½ × d × h₂
= ½ × d × (h₁ + h₂)
Part 4 — Cost on Land Area
Once the area is known, the total cost of any work (cultivating, fencing, purchasing) is found by:
Total Cost (T) = Area (A) × Rate per unit (R)
∴ T = A × R
A = T ÷ R (find area from cost)
R = T ÷ A (find rate from cost)
These four building blocks — triangle by base-height, Heron's formula, quadrilateral by diagonal, and the cost formula — are the complete set of tools needed to solve all problems in this chapter.