Surface Areas and Volumes

Surface Areas and Volumes

This chapter explores the formulas and concepts needed to calculate the surface areas and volumes of various three-dimensional (3D) shapes. Understanding these concepts helps in solving real-life problems involving containers, buildings, and other objects.

Key Concepts

• Surface Area: The total area that the surface of a 3D object occupies.
• Volume: The amount of space enclosed within a 3D object.
• Different shapes have different formulas for surface area and volume.

Important 3D Shapes and Their Formulas

Shape	Surface Area	Volume
Cuboid	2(lb + bh + hl)	l × b × h
Cube	6a2	a3
Cylinder	2πr(r + h)	πr2h
Cone	πr(l + r)	(1/3)πr2h
Sphere	4πr2	(4/3)πr3
Hemisphere	3πr2 (curved + base)	(2/3)πr3

Applications

• Finding the amount of material needed to make a box, can, or tent.
• Calculating the capacity of tanks, containers, and rooms.
• Solving real-life problems involving cost, painting, wrapping, or filling objects.

Sample Problems

1. Find the surface area and volume of a cube with side 5 cm.
   Solution:
   Surface area = 6 × 5² = 6 × 25 = 150 cm²
   Volume = 5³ = 125 cm³
2. A cylindrical tank has radius 7 m and height 3 m. Find its volume.
   Solution:
   Volume = π × 7² × 3 = π × 49 × 3 = 462π m³ ≈ 1451.33 m³
3. Find the total surface area of a sphere of radius 10 cm.
   Solution:
   Surface area = 4π × 10² = 4π × 100 = 400π cm² ≈ 1256.64 cm²

Tips

• Always use the same units for all dimensions.
• Remember to include units in your answers (cm² for area, cm³ for volume).
• Draw diagrams to visualize the problem.
• For composite shapes, break them into simpler parts.

Practice Exercises

1. Find the volume of a cuboid with length 8 cm, breadth 5 cm, and height 2 cm.
2. Calculate the curved surface area of a cylinder with radius 4 cm and height 10 cm.
3. A cone has a base radius of 3 cm and height 4 cm. Find its volume.
4. What is the surface area of a hemisphere with radius 6 cm?
5. How much water can a spherical tank of radius 2 m hold?

Summary

• Surface area is the total area covering a 3D object.
• Volume is the space occupied by a 3D object.
• Different shapes have different formulas for surface area and volume.
• Apply these concepts to solve real-world problems.