Quantitative Aptitude
Area and Perimeter (2D Mensuration); Volume and Surface Area (3D Mensuration)
C-CAT
Area and Perimeter (2D Mensuration)
21.1 Rectangle
| Property | Formula |
|---|---|
| Area | length × breadth |
| Perimeter | 2(l + b) |
| Diagonal | √(l² + b²) |
21.2 Square
| Property | Formula |
|---|---|
| Area | side² = (diagonal)²/2 |
| Perimeter | 4 × side |
| Diagonal | side × √2 |
21.3 Triangle
| Property | Formula |
|---|---|
| Area (base, height) | ½ × base × height |
| Area (Heron's) | √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2 |
| Equilateral triangle area | (√3/4) × side² |
| Area (two sides, angle) | ½ × a × b × sin(C) |
| Perimeter | a + b + c |
21.4 Circle
| Property | Formula |
|---|---|
| Area | πr² |
| Circumference | 2πr |
| Diameter | 2r |
| Arc length | 2πrθ/360 (θ in degrees) |
| Area of sector | πr²θ/360 |
| Area of segment | Area of sector - Area of triangle |
21.5 Parallelogram and Rhombus
| Shape | Area | Perimeter |
|---|---|---|
| Parallelogram | base × height | 2(a + b) |
| Rhombus | ½ × d₁ × d₂ | 4 × side |
| Trapezium | ½(sum of parallel sides) × height | Sum of all sides |
21.6 Room Area
Area of Four Walls of a Room:
Area = 2 × (length + breadth) × height
21.7 Semi-circle
| Property | Formula |
|---|---|
| Area | (π/2)r² |
| Perimeter | πr + 2r = r(π + 2) |
21.8 Important Relations
- If side of square is doubled → Area becomes 4 times
- If radius of circle is doubled → Area becomes 4 times
- Area ∝ r² (direct proportion with square of radius)
Example: If radius increased by 50%:
- New radius = 1.5r, New area = π(1.5r)² = 2.25πr²
- Increase = 2.25 - 1 = 1.25 = 125% increase in area
Volume and Surface Area (3D Mensuration)
22.1 Cuboid
| Property | Formula |
|---|---|
| Volume | l × b × h |
| Total Surface Area | 2(lb + bh + lh) |
| Lateral Surface Area | 2h(l + b) |
| Diagonal | √(l² + b² + h²) |
22.2 Cube (all sides = a)
| Property | Formula |
|---|---|
| Volume | a³ |
| Total Surface Area | 6a² |
| Lateral Surface Area | 4a² |
| Diagonal | a√3 |
22.3 Cylinder
| Property | Formula |
|---|---|
| Volume | πr²h |
| Curved Surface Area | 2πrh |
| Total Surface Area | 2πr(h + r) |
22.4 Cone
| Property | Formula |
|---|---|
| Slant height (l) | √(r² + h²) |
| Volume | (1/3)πr²h |
| Curved Surface Area | πrl |
| Total Surface Area | πr(l + r) |
22.5 Sphere
| Property | Formula |
|---|---|
| Volume | (4/3)πr³ |
| Surface Area | 4πr² |
22.6 Hemisphere
| Property | Formula |
|---|---|
| Volume | (2/3)πr³ |
| Curved Surface Area | 2πr² |
| Total Surface Area | 3πr² |
22.7 Important Tricks
Cube melted and recast:
- Volume remains constant
- a³ (old cube) = number × new shape volume
Converting a shape:
- If a cone/sphere is melted to make a cylinder:
- Volume of cone/sphere = Volume of cylinder
- Solve for unknown dimension
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