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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
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
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
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
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 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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