21. Sphere, Hollow Sphere, Prism & Pyramid Formulas
In this part, we learn the important mensuration formulas for Sphere, Hollow Sphere, Prism and Pyramid.
These formulas are useful for solving questions based on Volume, Curved Surface Area (CSA), Lateral Surface Area (LSA) and Total Surface Area (TSA).
The formulas in this section follow the corresponding Surface Area & Volume section of the provided Mensuration Formula PDF. :contentReference[oaicite:1]{index=1}
1. Sphere
A sphere is a three-dimensional solid in which every point on the surface is at the same distance from its centre.
Let r be the radius of the sphere.
Sphere Formulas
The PDF gives the sphere formulas as volume 4/3 πr³ and surface area 4πr². :contentReference[oaicite:2]{index=2}
Example 1: Volume of a Sphere
Find the volume of a sphere whose radius is 7 cm. Take π = 22/7.
Example 2: Surface Area of a Sphere
Find the surface area of a sphere whose radius is 7 cm. Take π = 22/7.
2. Hollow Sphere
A hollow sphere has an outer spherical surface and an inner spherical surface.
Let:
- R = Outer radius
- r = Inner radius
The material occupies the space between the outer and inner spherical surfaces.
Hollow Sphere Formulas
For a hollow sphere, the volume of material is found by subtracting the volume of the inner sphere from the volume of the outer sphere.
Example 3: Hollow Sphere
A hollow sphere has outer radius 7 cm and inner radius 3 cm. Find the volume of material. Take π = 22/7.
3. Prism
A prism is a three-dimensional solid having two equal and parallel bases connected by rectangular or parallelogram-shaped side faces.
Let:
- Base Area = Area of one base
- Base Perimeter = Perimeter of the base
- h = Height of prism
Prism Formulas
These are the general prism formulas shown in the provided formula sheet. :contentReference[oaicite:3]{index=3}
Example 4: Volume of a Prism
A prism has a base area of 30 cm² and height 10 cm. Find its volume.
Example 5: LSA of a Prism
The perimeter of the base of a prism is 24 cm and its height is 8 cm. Find its LSA.
4. Pyramid
A pyramid is a solid with a polygonal base and triangular faces that meet at one common point called the vertex.
Let:
- Base Area = Area of the base
- Base Perimeter = Perimeter of the base
- h = Vertical height
- l = Slant height
Pyramid Formulas
The formula sheet gives the pyramid volume as one-third of base area × height, LSA as one-half of base perimeter × slant height, and TSA as LSA + base area. :contentReference[oaicite:4]{index=4}
Example 6: Volume of a Pyramid
A pyramid has a base area of 60 cm² and height 12 cm. Find its volume.
Example 7: LSA of a Pyramid
A pyramid has a base perimeter of 40 cm and slant height of 9 cm. Find its LSA.
5. Quick Formula Table
| Solid | Volume | CSA / LSA | TSA / Surface Area |
|---|---|---|---|
| Sphere | ⁴⁄₃πr³ | — | 4πr² |
| Hollow Sphere | ⁴⁄₃π(R³ − r³) | — | 4π(R² + r²) |
| Prism | Base Area × Height | Base Perimeter × Height | LSA + 2 × Base Area |
| Pyramid | ⅓ × Base Area × Height | ½ × Base Perimeter × Slant Height | LSA + Base Area |
6. Important Symbols
| Symbol | Meaning |
|---|---|
| r | Radius |
| R | Outer radius of a hollow sphere |
| h | Height |
| l | Slant height |
| π | Pi, generally taken as 22/7 or 3.14 |
| LSA | Lateral Surface Area |
| CSA | Curved Surface Area |
| TSA | Total Surface Area |
7. Important Difference: Sphere vs Hollow Sphere
Sphere: A solid sphere has only one outer surface.
Hollow Sphere: A hollow sphere has both an outer and an inner spherical surface.
For a hollow sphere, the material volume is obtained by subtracting the inner sphere volume from the outer sphere volume.
8. Prism vs Pyramid
| Prism | Pyramid |
|---|---|
| Has two equal and parallel bases. | Has one base and triangular faces. |
| Volume = Base Area × Height | Volume = ⅓ × Base Area × Height |
| LSA = Base Perimeter × Height | LSA = ½ × Base Perimeter × Slant Height |
| TSA = LSA + 2 × Base Area | TSA = LSA + Base Area |
9. Practice Questions
1. Find the volume of a sphere with radius 7 cm. Use π = 22/7.
2. Find the surface area of a sphere with radius 14 cm. Use π = 22/7.
3. A hollow sphere has outer radius 7 cm and inner radius 3 cm. Find the volume of material.
4. A prism has a base area of 45 cm² and height 8 cm. Find its volume.
5. The base perimeter of a prism is 30 cm and its height is 12 cm. Find its LSA.
6. A pyramid has a base area of 72 cm² and height 15 cm. Find its volume.
7. A pyramid has a base perimeter of 36 cm and slant height 10 cm. Find its LSA.
10. Answers
| Question | Answer |
|---|---|
| 1 | 1437.33 cm³ approximately |
| 2 | 2464 cm² |
| 3 | Approximately 1323.81 cm³ |
| 4 | 360 cm³ |
| 5 | 360 cm² |
| 6 | 360 cm³ |
| 7 | 180 cm² |
11. Exam Shortcuts
Sphere:
Hollow Sphere:
Prism:
Pyramid:
Remember that the volume of a pyramid is one-third of the product of base area and height.
12. Common Mistakes to Avoid
1. Do not use diameter in place of radius.
2. Remember that volume is measured in cubic units.
3. Surface area is measured in square units.
4. For a hollow sphere, use both the outer radius and inner radius.
5. Do not forget the factor 1/3 in the volume of a pyramid.
6. Do not confuse the vertical height h with the slant height l.
13. One-Minute Revision
Source: Provided Mensuration Formula PDF, Surface Area & Volume section. The source page specifically lists Sphere, Hollow Sphere, Prism and Pyramid formulas. :contentReference[oaicite:5]{index=5}