7 min readConcept sheetUsually a Part III question (5 marks)

Area of a quadrilateral

Split a quadrilateral into two triangles, or use the column method with all four vertices taken anticlockwise.

Area of quadrilateral ABCD(1/2){(x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁) − (x₂y₁ + x₃y₂ + x₄y₃ + x₁y₄)} sq. units

Order of verticesA, B, C, D in order around the shape, anticlockwise

Other methodArea ABCD = area of triangle ABC + area of triangle ACD

Order matters

A quadrilateral is a shape with four sides and four vertices A, B, C and D. Go round the shape in order: A to B to C to D and back to A. Never jump across a diagonal (the line joining opposite corners). Take the vertices anticlockwise (counter-clockwise), as your textbook asks. Plot the points first, so you can see the right order.

Column method for four vertices

1
Plot the points and name them A, B, C, D anticlockwise.
2
Write the points in two columns (x and y). Repeat A at the bottom.
3
First sum: each x times the y in the row below: x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁.
4
Second sum: each y times the x in the row below: y₁x₂ + y₂x₃ + y₃x₄ + y₄x₁.
5★
Area = (1/2)(first sum − second sum) sq. units.

★ key step

Example 1: area from four vertices

Problem

Find the area of the quadrilateral with vertices A(−3, −2), B(4, −3), C(5, 3) and D(−2, 4).

Working

Formula: Area = (1/2){(x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁) − (x₂y₁ + x₃y₂ + x₄y₃ + x₁y₄)}
Plot the points: A is bottom left, B bottom right, C top right, D top left. So A → B → C → D is anticlockwise.
First sum: x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁ = (−3)(−3) + (4)(3) + (5)(4) + (−2)(−2)
= 9 + 12 + 20 + 4 = 45
Second sum: x₂y₁ + x₃y₂ + x₄y₃ + x₁y₄ = (4)(−2) + (5)(−3) + (−2)(3) + (−3)(4)
= −8 − 15 − 6 − 12 = −41
Area = (1/2){45 − (−41)} = (1/2)(86) = 43

Answer

Area of ABCD = 43 sq. units

Example 2: a paddy field near Thanjavur

Problem

A farmer's paddy field has corners P(1, 1), Q(9, 2), R(8, 7) and S(2, 6) on a map where 1 unit = 10 m. Find the area of the field in square metres.

Working

Use the same formula. P, Q, R, S go anticlockwise (bottom left, bottom right, top right, top left).
First sum: (1)(2) + (9)(7) + (8)(6) + (2)(1) = 2 + 63 + 48 + 2 = 115
Second sum: (9)(1) + (8)(2) + (2)(7) + (1)(6) = 9 + 16 + 14 + 6 = 45
Area = (1/2)(115 − 45) = (1/2)(70) = 35 sq. units
1 unit = 10 m, so 1 sq. unit = 10 m × 10 m = 100 m²
Area = 35 × 100 = 3500 m²

Answer

The field has an area of 3500 m².

Common mistakes

Taking the vertices out of order (for example A, C, B, D). This gives a wrong area.
Forgetting the products that close the loop: x₄y₁ in the first sum and x₁y₄ in the second.
Forgetting to change square units into m² when 1 unit is not 1 m.
Dropping a minus sign when a coordinate is negative.

In real life

House plots, classroom floors and kolam patterns are often four-sided. If you know the corner points on a plan, this formula gives the area, for example to order the right number of floor tiles.

Exam angle

2 marks: “Find the area of the quadrilateral with vertices (0, 0), (4, 0), (4, 3) and (0, 3).” First sum = 0 + 12 + 12 + 0 = 24 and second sum = 0. Area = (1/2)(24) = **12 sq. units**. (It is a 4 by 3 rectangle.)

5 marks: “Find the area of the quadrilateral whose vertices are A(−4, −3), B(3, −1), C(3, 2) and D(−2, 4).” The points go anticlockwise. First sum = 4 + 6 + 12 + 6 = 28. Second sum = −9 − 3 − 4 − 16 = −32. Area = (1/2)(28 + 32) = **30 sq. units**.

Remember it

Go round the shape, not across it: A, B, C, D and back to A.