NCERT Notes, Formulas, Solved Examples & Interactive 3D Shape Calculators
A cone is a 3D shape generated by rotating a right-angled triangle around one of its perpendicular sides. Think of an ice-cream cone or a birthday hat!
Net of a Cone
A cylinder has two circular bases and one curved surface. Example: water tank, drum, pipe.
Input Radius (r) and Height (h) to find everything!
A sphere is the set of all points in 3D space equidistant from a center point.
Sphere Area = \(4\pi r^2\)
| Shape | CSA / LSA | Total Surface Area | Volume |
|---|---|---|---|
| Cone | \( \pi r l \) | \( \pi r (l + r) \) | \( \frac{1}{3} \pi r^2 h \) |
| Sphere | \( 4 \pi r^2 \) | \( 4 \pi r^2 \) | \( \frac{4}{3} \pi r^3 \) |
| Hemisphere | \( 2 \pi r^2 \) | \( 3 \pi r^2 \) | \( \frac{2}{3} \pi r^3 \) |
| Solid | TSA | Volume |
|---|---|---|
| Cuboid | \(2(lb + bh + hl)\) | \(l \times b \times h\) |
| Cube | \(6a^2\) | \(a^3\) |
Volume represents the space occupied, while Capacity is the amount of liquid a container can hold.
Convert Volume (cm³) to Liters
| Property | Sphere | Hemisphere |
|---|---|---|
| CSA | \(4\pi r^2\) | \(2\pi r^2\) |
| TSA | \(4\pi r^2\) | \(3\pi r^2\) |
| Volume | \( \frac{4}{3}\pi r^3 \) | \( \frac{2}{3}\pi r^3 \) |
A water tank is in the shape of a cylinder surmounted by a hemisphere. Radius = 7 m, height of cylindrical part = 10 m.
Assertion (A): Volume of a cone is one-third the volume of a cylinder.
Reason (R): Both have same base radius and height.
✔ Both A and R are true and R explains A
Common board-style questions. Click to reveal simplified solutions.
Formula: CSA = πrl
= (22/7) × 7 × 10
= 220 cm²
Formula: TSA of hemisphere = 3πr²
= 3 × (22/7) × (3.5)²
= 3 × (22/7) × 12.25 = 115.5 cm²
Formula: V = πr²h
= (22/7) × (1.4)² × 2
= (22/7) × 1.96 × 2 = 12.32 m³
1 m³ = 1000 litres → 12320 litres
Volume: V = (4/3)πr³
New V = (4/3)π(2r)³ = (4/3)π × 8r³ = 8V
∴ Volume becomes 8 times
Key: Volume scales as the cube of the linear dimension.