Mensuration 2D

Module 17 — Lesson 0017 · In Depth · 40 min

2D mensuration measures flat shapes — boundary (perimeter) and space occupied (area). Every formula is either counting squares (area) or walking the edge (perimeter). Learn the general case, then the shortcuts for regular shapes.

1. The Fundamental Shapes

a) Square — side a

• Area: a²
• Perimeter: 4a
• Diagonal: a√2 — by Pythagoras on half-square.

b) Rectangle — length l, width b

• Area: l × b
• Perimeter: 2(l + b)
• Diagonal: √(l² + b²) — Pythagoras on half-rectangle.

c) Triangle

• General: Area = (1/2) × base × height (height is perpendicular to that base).
• Heron's formula (all sides a, b, c known):
  s = (a+b+c)/2 (semi-perimeter), Area = √[s(s−a)(s−b)(s−c)]
  Use when height is not given — works for any triangle.
• Equilateral (side a): Area = (√3/4) a², Height = (√3/2) a. Derive from Pythagoras on half-equilateral (30-60-90 triangle).

d) Circle — radius r, diameter d = 2r

• Area: πr²
• Circumference (perimeter): 2πr = πd
• Use π ≈ 22/7 when r is a multiple of 7; otherwise 3.14.

2. Advanced Shapes

a) Parallelogram

Area: base × height (height is perpendicular distance between bases, not side length). Perimeter: 2 × (sum of adjacent sides).

b) Trapezium — parallel sides a, b, height h

Area: (1/2) × (a + b) × h — average of parallel sides times distance between them. Think of it as "trapezium is half a parallelogram".

c) Regular Polygon — n sides of length s

• Perimeter: n × s
• Hexagon area: (3√3/2) s² — 6 equilateral triangles, same as in Geometry (Module 23).
• General area: (1/2) × perimeter × apothem (apothem = distance from center to midpoint of a side). Useful when apothem is given.

3. Circle Properties — Sectors and Segments

a) Sector (Slice of Pizza)

Part enclosed by two radii and an arc, central angle θ:

• Area: (θ/360) × πr²
• Arc length: (θ/360) × 2πr

b) Chord and Segment

• Chord length: 2r sin(θ/2)
• Area of segment (minor) = area of sector − area of triangle formed by radii and chord. Triangle area = (1/2) r² sin θ, so segment = (θ/360)πr² − (1/2)r² sin θ.

4. Inscribed and Circumscribed Figures

a) Rectangle in a Circle

Diagonal of rectangle = diameter of circle: l² + b² = (2r)². Every corner of the rectangle touches the circle — so the diagonal is a diameter (Thales' theorem: angle in a semicircle is 90°).

b) Circle in Square vs Square in Circle

c) Triangle Circles

• Inradius r (incircle): r = Area / s where s = semi-perimeter. Derivation: area = sum of three triangles with height r.
• Circumradius R (circumcircle): R = (a×b×c) / (4×Area)
• Right triangle shortcuts: Inradius = (P + B − H)/2, Circumradius = H/2 (hypotenuse is diameter).

5. Pathways and Borders

a) Path Outside a Rectangle

Rectangle l × b, path width w outside:

• Outer length: l + 2w, Outer width: b + 2w (w on both sides)
• Area of path = Area(outer) − Area(inner) = (l+2w)(b+2w) − lb
• Quick: = 2w(l + b + 2w)

b) Path Inside a Rectangle

• Inner length: l − 2w, Inner width: b − 2w
• Area of path = lb − (l−2w)(b−2w) = 2w(l + b − 2w)

Pitfall: Outside adds 2w, inside subtracts 2w — but students forget the factor 2 (w on both sides). A 2m border around a rectangle adds 4m to total length.

6. Summary of Variables

Notes

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