Surface Area, Volume And Capacity
Surface Area, Volume, and Capacity: Comprehensive Guide
1. Surface Area
Surface area is the total area of all faces or surfaces of a three-dimensional object.
Common Shapes and Their Surface Area Formulas:
Cube
Surface Area = 6s², where s = length of side
Rectangular Prism
Surface Area = 2(lw + lh + wh), where:
- l = length
- w = width
- h = height
Sphere
Surface Area = 4πr², where r = radius
Cylinder
Surface Area = 2πr² + 2πrh, where:
- r = radius of base
- h = height
2. Volume
Volume is the amount of three-dimensional space enclosed by a closed surface.
Cube
Volume = s³, where s = length of side
Rectangular Prism
Volume = l × w × h
Sphere
Volume = (4/3)πr³
Cylinder
Volume = πr²h
3. Capacity
Capacity refers to the maximum amount that something can contain, usually measured in units of volume.
Common Conversions:
- 1 m³ = 1000 liters
- 1 liter = 1000 ml
- 1 cm³ = 1 ml
4. Practical Applications
Surface area calculations are used in:
- Painting and decorating
- Construction materials estimation
- Heat transfer calculations
Volume calculations are essential for:
- Storage container design
- Shipping and packaging
- Construction and engineering
Capacity measurements are crucial in:
- Food and beverage industry
- Chemical processing
- Fuel storage
5. Example Problems and Solutions
Problem 1: Surface Area of a Gift Box
A rectangular gift box has a length of 30 cm, width of 20 cm, and height of 15 cm. Calculate:
- The surface area of wrapping paper needed
- The amount of ribbon needed to wrap around the box once vertically and once horizontally, plus a 20 cm bow
Solution:
a) Surface Area calculation:
- Surface Area = 2(lw + lh + wh)
- = 2(30×20 + 30×15 + 20×15)
- = 2(600 + 450 + 300)
- = 2(1350)
- = 2700 cm²
b) Ribbon length calculation:
- Vertical wrap = 2(height + width) = 2(15 + 20) = 70 cm
- Horizontal wrap = 2(height + length) = 2(15 + 30) = 90 cm
- Bow = 20 cm
- Total ribbon needed = 70 + 90 + 20 = 180 cm
Problem 2: Swimming Pool Volume
A swimming pool is 25 meters long, 10 meters wide, and has a depth that gradually increases from 1.2 meters to 2.8 meters. Calculate:
- The volume of water needed to fill the pool
- The cost of filling the pool if water costs $1.50 per 1000 liters
Solution:
a) Volume calculation:
- Average depth = (1.2 + 2.8) ÷ 2 = 2 meters
- Volume = length × width × average depth
- = 25 × 10 × 2
- = 500 cubic meters
- = 500,000 liters (since 1 m³ = 1000 liters)
b) Cost calculation:
- Cost per 1000 liters = $1.50
- Number of thousands of liters = 500,000 ÷ 1000 = 500
- Total cost = 500 × $1.50 = $750
Problem 3: Paint Required for a Cylindrical Water Tank
A cylindrical water tank has a radius of 3 meters and a height of 8 meters. Calculate:
- The surface area to be painted (including top and bottom)
- The amount of paint needed if 1 liter covers 10 m²
Solution:
a) Surface area calculation:
- Lateral surface area = 2πrh = 2 × π × 3 × 8 = 150.80 m²
- Area of top and bottom circles = 2πr² = 2 × π × 3² = 56.55 m²
- Total surface area = 150.80 + 56.55 = 207.35 m²
b) Paint required:
- Paint needed = Total surface area ÷ Coverage per liter
- = 207.35 ÷ 10
- = 20.74 liters
- Round up to 21 liters to ensure complete coverage
