Quantitative Aptitude - Mensuration
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The document defines the formulas for calculating the area, perimeter, volume, and surface area of various geometric shapes including triangles, rectangles, squares, parallelograms, rhombuses, trapezoids, kites, circles, sectors, segments, cubes, cuboids, cylinders, cones, hemispheres, spheres, prisms, pyramids, frustums of cones and pyramids. For each shape, the key formulas for area, perimeter, volume, or surface area are provided.
Quantitative Aptitude - Mensuration
- 2. • Area = ½ (a x h1) = ½(b x h2) = ½(c x h3) • Semi-Perimeter s = (a + b + c)/2 • Area = √(s x (s – a) x (s - b) x (s - c)) • Perimeter = a + b + c EQUILATERAL TRIANGLE (Sides a) • Area = √3/4 x a2 • Perimeter = 3 x a • Height = √3/2 x a TRIANGLE (Sides a, b, c; Opp. Angles A, B, C respectively; heights h1, h2 and h3 respectively) a a a A B C A aB b h1 h2 h3 c C
- 3. RECTANGLE (length l; breadth b) • Area = l x b • Perimeter = 2(l + b) SQUARE (side a) • Area = a2 • Perimeter = 4a l b a a
- 4. PARALLELOGRAM (height h, base b, other side a) • Area = b x h • Perimeter = 2 x (a + b) • Area = ½ (d1 x d2) • Perimeter = 4 x a h d1 d2 b a a RHOMBUS (side a; diagonals of length d1 and d2 ) a
- 5. • Area = ½ (a + b) x h • Perimeter = a + b + c + d • Area = ½ (d1 x d2) • Perimeter = a + b + c + d a b h d1 d2 TRAPEZIUM (distance between parallel sides h, length of parallel sides a & b) KITE (diagonals of length d1 and d2 ) c d dc b a
- 6. • Area = πr2/2 • Perimeter= πr + 2r • Area = πr2 • Circumference = 2πr r rO O CIRCLE (radius r) SEMI-CIRCLE (radius r)
- 7. • Area = /360 x (πr2) • Length of arc = ( /360 x 2πr) • Perimeter = ( /360 x 2πr) + 2r SECTOR (centre O, radius r, angle of sector ) SEGMENT (centre O, radius r, angle of sector ) • Area = 1/2r2 x {( /180) x π –Sin } r O r O
- 8. • Volume = a3 • Total Surface Area = 6a2 • Longest diagonal = a x √3 • Volume = l x b x h • Total Surface Area = 2{(l x b) + (b x h) + (l x h)} • Longest diagonal = √(l2 + b2 + h2) a a a l b h CUBE (length = breadth = height = a) CUBOID (length l; breadth b; height h)
- 9. • Volume = π x r2 x h • Total Surface Area = 2π x r x (h + r) • Curved (Lateral) Surface Area = 2π x r x h • Volume = 1/3 π x r2 xh • Total Surface Area = π x r x (l + r) • Curved (Lateral) Surface Area = π x r x l r r h h r l CYLINDER (height h; radius of base r) CONE (height h; radius of base r; slant height l )
- 10. • Volume = 4/3 π x r3 • Total Surface Area = 4π x r2 • Volume = 2/3 π x r3 • Total Surface Area = 3π x r2 • Curved (Lateral) Surface Area = 2π x r2 r r HEMISPHERE (radius r) SPHERE (radius r)
- 11. PRISM (length of side of base a; number of sides of base n; height h) • For a right prism, base can be any shape like square, triangle, pentagon or hexagon as shown • Volume = area of base x h • Total Surface Area = 2 x area of base + n x (h x a) • Lateral surface area = n x (a x h) PYRAMID (length of side of base a; number of sides of base n; height h; slant height l) • For a right pyramid, base can be any shape like square, triangle, pentagon or hexagon as shown • Volume = 1/3 x area of base x h • Lateral surface area = ½ x (n x a) x l • Total Surface Area = area of base + lateral surface area ha l n = 6 for hexagon a n = 6 for hexagon h
- 12. FRUSTUM OF CONE Slant height l; height h, radius of small base r; radius of large base R • Volume = 1/3π x h (R2 + r2 + Rr) • Total Surface Area = π x (R2 + r2 + Rl + rl) • Lateral Surface Area = π x l (R + r) • Slant height l = √{(R-r)2 + h2} FRUSTUM OF PYRAMID Slant height l; height h, side of base a • Volume = 1/3 x h x (A1 + A2 + √A1A2) • Lateral Surface Area = ½ x perimeter of base x l • Total Surface Area = LSA + A1 + A2 h l R r l A2 a h A1