Introduction & Key Definitions
Mensuration is the study of measuring lengths, areas, and volumes of geometric figures and solids. Below are the key definitions you need:
- 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.
Plane Figures — lengths & areas
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)
Find the area of a triangle with base 10 cm and height 6 cm.
Find the perimeter of a rectangle of length 8 cm and breadth 5 cm.
Arcs & Chords
Arc (deg): 2πr × (θ/360) — Arc (rad): r × θ
Chord = 2r × sin(θ/2)
Sectors & Segments
Sector area = ½ r² θ (radians) or π r² × (θ/360). Segment area = Sector − Triangle area.
Surface Areas & Volumes
Cuboid TSA = 2(lb + bh + lh) V = l × b × h
Cylinder TSA = 2πr(r + h) V = π r² h
Cone TSA = π r (r + l) V = ⅓ π r² h
Sphere TSA = 4π r² V = 4/3 π r³
Composite Figures & Earth Distances
Break complex shapes into known solids and sum/subtract volumes or areas as needed.
Great-circle distance (Haversine)
d = 2R · arcsin(√[hav(φ₂−φ₁) + cosφ₁ cosφ₂ · hav(λ₂−λ₁)]) where hav(θ)=sin²(θ/2)
Downloads
Editable and printable resources for study.
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