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Mensuration E‑Book

Edwin Ogie Library: Mensuration E‑Book

Edwin Ogie Library

Mensuration E‑Book

Objectives:
• Calculate perimeters and areas of plane figures;
• Find lengths of arcs, chords, sectors & segments;
• Determine surface areas & volumes of solids & composites;
• Compute distances on Earth’s surface using latitudes & longitudes.

Page 1: Introduction & Key Definitions

Mensuration is the study of measuring lengths, areas, and volumes of geometric figures and solids.

  • Perimeter: total distance around a plane figure.
  • Area: measure of the region enclosed by a plane figure.
  • Surface Area: sum of the areas of all faces of a solid.
  • Volume: space occupied by a solid.
  • Arc Length: portion of a circle’s circumference.
  • Chord: straight line joining two points on a circle.
  • Sector: region between two radii and the included arc.
  • Segment: region between a chord and its arc.

Page 2: Lengths & Areas of Plane Figures

Triangle

Perimeter = a + b + c
Area = ½ × base × height

Quadrilateral

  • Rectangle: P=2(l+b), A=l×b
  • Square: P=4a, A=a²
  • Parallelogram: P=2(a+b), A=base×height
  • Trapezium: P=sum of sides, A=½(h)(sum of parallel sides)

Example 1:

Find the area of a triangle with base 10 cm and height 6 cm.

Area = ½×10×6 = 30 cm²

Example 2:

Find the perimeter of a rectangle of length 8 cm and breadth 5 cm.

P = 2(8+5) = 26 cm

Page 3: Lengths of Arcs & Chords

Arc Length (degrees) = 2πr×(θ/360)
Arc Length (radians) = r×θ

Chord Length = 2r × sin(θ/2)

Example 3:

Find the length of an arc of radius 7 cm subtending 60°.

Arc = 2π×7×(60/360) = (7π/3) ≈ 7.33 cm

Example 4:

Find the chord length for the same circle and angle.

Chord = 2×7×sin(30°) = 14×0.5 = 7 cm

Page 4: Perimeters & Areas of Sectors & Segments

Sector Area = ½r²θ (radians) or πr²×(θ/360)

Segment Area = Sector Area – Triangle Area

Sector Segment

Example 5:

Compute the area of a sector with r=5 cm, θ=90°.

Sector A = π×5²×(90/360) = (25π/4) ≈19.63 cm²

Example 6:

Find the corresponding segment area.

Triangle A = ½×5×5×sin90° = 12.5; Segment A = 19.63–12.5 ≈7.13 cm²

Page 5: Surface Areas & Volumes of Solids

Simple Solids

Cuboid:

TSA=2(lb+bh+lh), V=l×b×h

Cylinder:

TSA=2πr(r+h), V=πr²h

Cone:

TSA=πr(r+l), V=⅓πr²h

Sphere:

TSA=4πr², V=4/3πr³

Page 6: Composite Figures

Composite Figure

Break into known solids, compute individual SA/volumes, then sum or subtract appropriately.

Example 7:

Find volume of a cylinder of radius 3 cm with a hemispherical cap.

V=πr²h + (2/3)πr³ = π×9h + (2/3)π×27.

Page 7: The Earth as a Sphere

Longitudes & Latitudes

Great-Circle Distance (Haversine):

d=2R·arcsin(√[hav(φ2−φ1)+cosφ1·cosφ2·hav(λ2−λ1)])

where hav(θ)=sin²(θ/2).

Example 8:

Compute distance between (φ1,λ1)=(0°,0°) and (φ2,λ2)=(0°,90°) on Earth radius 6371 km.

d=2·6371·arcsin(1/√2)=10007.5 km (quarter circumference).

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JAMB Mensuration CBT Quiz – 20 Questions

JAMB Mensuration CBT Quiz

20 questions drawn from key mensuration formulas: plane figures, arcs, sectors, solids, composites & Earth distances. You have 10 minutes. Click “Start Quiz” to begin after a 10‑second delay.

Explanations

Q1–Q2:

Triangle P = a+b+c; A = ½·b·h

Q3–Q4:

Rectangle P = 2(l+b); A = l·b

Q5–Q6:

Square P = 4a; A = a²

Q7:

Parallelogram A = b·h

Q8:

Trapezium A = ½·h·(sum of parallels)

Q9–Q10:

Arc = 2πr·θ/360 or r·θ (radians)

Q11:

Chord = 2r·sin(θ/2)

Q12–Q13:

Sector A = ½r²θ or πr²·θ/360

Q14:

Segment A = Sector A – Triangle A

Q15:

Cuboid TSA = 2(lb+bh+lh)

Q16:

Cylinder V = πr²h

Q17:

Cone TSA = πr(r+l)

Q18:

Sphere V = 4/3πr³

Q19:

Composite: V = πr²h + 2/3πr³

Q20:

Quarter circumference = 2πR/4

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