Solved Exercises on Surface Area of Cylinders in Grade 8

Master surface area of cylinders: radius, height, diameter calculations, and real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Cylinder Surface Area
Exercise 1
Find the surface area of a cylinder with radius 5 cm and height 12 cm. Use π ≈ 3.14.
Definition:

Surface Area of Cylinder: The total area of all surfaces of a cylinder. The formula is SA = 2πr² + 2πrh, where r is the radius of the circular base and h is the height. This includes both circular bases and the curved lateral surface.

Surface Area Calculation Method:
  1. Identify the radius (r) and height (h)
  2. Apply the formula SA = 2πr² + 2πrh
  3. Calculate the area of both circular bases: 2πr²
  4. Calculate the lateral surface area: 2πrh
  5. Add the two areas together
  6. Include the correct units (square units)
Given Values
r = 5 cm, h = 12 cm
Formula
SA = 2πr² + 2πrh
Result
SA = 533.8 cm²
Step 1: Write down the formula

SA = 2πr² + 2πrh

Step 2: Calculate area of both circular bases

2πr² = 2 × π × (5)² = 2 × π × 25 = 50π

Using π ≈ 3.14: 50π ≈ 50 × 3.14 = 157 cm²

Step 3: Calculate the lateral surface area

2πrh = 2 × π × 5 × 12 = 120π

Using π ≈ 3.14: 120π ≈ 120 × 3.14 = 376.8 cm²

Step 4: Add both areas

SA = 157 + 376.8 = 533.8 cm²

SA = 533.8 cm²
Final answer:

The surface area of the cylinder is 533.8 square centimeters.

Applied rules:

• Surface Area Formula: SA = 2πr² + 2πrh

• Units: Surface area is measured in square units

• Components: Includes both circular bases and lateral surface

2 Diameter to Radius Conversion
Exercise 2
A cylindrical water tank has a diameter of 8 feet and a height of 10 feet. Find its surface area. Use π ≈ 3.14.
Definition:

Diameter and Radius Relationship: The diameter (d) is twice the radius (r), so d = 2r or r = d/2. Always convert diameter to radius before using the surface area formula.

Given Values
d = 8 ft, h = 10 ft
Convert to Radius
r = 4 ft
Surface Area
SA = 351.68 ft²
Step 1: Convert diameter to radius

r = d/2 = 8/2 = 4 feet

Step 2: Apply the surface area formula

SA = 2πr² + 2πrh = 2π(4)² + 2π(4)(10)

Step 3: Calculate each component

Area of bases: 2πr² = 2 × π × 16 = 32π ≈ 32 × 3.14 = 100.48 ft²

Lateral surface: 2πrh = 2 × π × 4 × 10 = 80π ≈ 80 × 3.14 = 251.2 ft²

Step 4: Add both areas

SA = 100.48 + 251.2 = 351.68 ft²

SA = 351.68 ft²
Final answer:

The surface area of the water tank is 351.68 square feet.

Applied rules:

• Diameter to Radius: r = d/2

• Surface Area Formula: SA = 2πr² + 2πrh

• Unit Consistency: Keep all measurements in the same units

3 Lateral Surface Area Only
Exercise 3
Find the lateral surface area of a cylinder with radius 6 inches and height 15 inches. Use π ≈ 3.14.
Definition:

Lateral Surface Area: The area of just the curved side of the cylinder, excluding the top and bottom circular bases. The formula is LSA = 2πrh.

Given Values
r = 6 in, h = 15 in
Formula
LSA = 2πrh
Lateral Surface Area
LSA = 565.2 in²
Step 1: Write down the lateral surface area formula

LSA = 2πrh

Step 2: Substitute the known values

LSA = 2 × π × 6 × 15

Step 3: Calculate

LSA = 2 × 3.14 × 6 × 15

LSA = 2 × 3.14 × 90

LSA = 565.2 in²

LSA = 565.2 in²
Final answer:

The lateral surface area of the cylinder is 565.2 square inches.

Applied rules:

• Lateral Surface Area: LSA = 2πrh (only curved side)

• Units: Surface area is measured in square units

• Application: Used when only the curved surface needs to be covered

Rules and methods, laws,...
\(SA = 2\pi r^2 + 2\pi rh\)
Surface Area of Cylinder
Total Surface Area
SA = 2πr² + 2πrh
Includes bases and lateral surface
Lateral Surface Area
LSA = 2πrh
Only curved side
Base Area
A = πr²
Area of one circular base
Diameter to Radius
r = d/2
Always convert first
Alternative Formula
SA = 2πr(r + h)
Factored form
Components: 2 circular bases + 1 rectangular lateral surface (curled)
Net of Cylinder: Rectangle (lateral) + 2 circles (bases)
Key definitions:

Cylinder: A three-dimensional shape with two parallel circular bases connected by a curved surface.

Radius (r): The distance from the center of the circular base to its edge.

Diameter (d): The distance across the circular base, passing through the center. d = 2r.

Height (h): The perpendicular distance between the two circular bases.

Surface Area: The total area of all surfaces of a three-dimensional object, measured in square units.

Complete methodology:
  1. Identify given information: Determine which measurements are provided
  2. Convert units if needed: Ensure all measurements are in the same units
  3. Apply the correct formula: Use SA = 2πr² + 2πrh
  4. Calculate components: Find area of bases and lateral surface separately
  5. Sum components: Add all areas together
  6. Include units: Always express the answer in square units
Tip 1: Always convert diameter to radius before using the formula.
Tip 2: Remember that surface area is always expressed in square units.
Tip 3: Be careful to distinguish between total surface area and lateral surface area.

Solution: Exercises 4 to 5
4 Real-World Application
Exercise 4
A cylindrical soup can has a diameter of 7 cm and a height of 10 cm. If the label covers only the lateral surface, how much area does the label cover? Use π ≈ 3.14.
Definition:

Real-World Application: Many practical situations require calculating only the lateral surface area, such as labels, paint coverage, or wrapping paper for the curved part of a cylinder.

Given Values
d = 7 cm, h = 10 cm
Convert to Radius
r = 3.5 cm
Label Area
LSA = 219.8 cm²
Step 1: Find the radius

r = d/2 = 7/2 = 3.5 cm

Step 2: Calculate the lateral surface area

LSA = 2πrh = 2 × 3.14 × 3.5 × 10

LSA = 2 × 3.14 × 35

LSA = 219.8 cm²

Step 3: Interpret the result

The label covers 219.8 square centimeters of the can's surface.

219.8 cm²
Final answer:

The label covers 219.8 square centimeters of the can's surface.

Applied rules:

• Lateral Surface Area: LSA = 2πrh for curved surface only

• Diameter to Radius: r = d/2

• Real-World Context: Labels cover lateral surface area

5 Comparative Analysis
Exercise 5
Cylinder A has radius 4 cm and height 6 cm. Cylinder B has radius 6 cm and height 4 cm. Which cylinder has the greater surface area, and by how much? Use π ≈ 3.14.
Definition:

Comparative Analysis: Calculating surface areas of different cylinders to compare their values. Note that changing dimensions affects surface area differently than volume.

Cylinder A
SA_A = 251.2 cm²
Cylinder B
SA_B = 376.8 cm²
Difference
ΔSA = 125.6 cm²
Step 1: Calculate surface area of Cylinder A

SA_A = 2πr² + 2πrh = 2π(4)² + 2π(4)(6)

SA_A = 2π(16) + 2π(24) = 32π + 48π = 80π

SA_A = 80 × 3.14 = 251.2 cm²

Step 2: Calculate surface area of Cylinder B

SA_B = 2πr² + 2πrh = 2π(6)² + 2π(6)(4)

SA_B = 2π(36) + 2π(24) = 72π + 48π = 120π

SA_B = 120 × 3.14 = 376.8 cm²

Step 3: Compare surface areas

SA_B > SA_A

Difference = SA_B - SA_A = 376.8 - 251.2 = 125.6 cm²

Step 4: Analyze the result

Cylinder B has the greater surface area. Even though A and B have swapped dimensions (4×6 vs 6×4), Cylinder B has the greater surface area because the radius affects both the base areas (which depend on r²) and the lateral surface area (which depends on r).

Cylinder B has greater surface area by 125.6 cm²
Final answer:

Cylinder B has the greater surface area by 125.6 square centimeters.

Applied rules:

• Surface Area Formula: SA = 2πr² + 2πrh for both cylinders

• Comparative Analysis: Calculate each surface area separately

• Radius Effect: Since bases depend on r², radius has significant impact

Surface Area of Cylinders Laws, Methods, and Properties
\(SA = 2\pi r^2 + 2\pi rh\)
Surface Area of Cylinder
Key definitions:

Surface Area of Cylinder: The measure of the total area of all surfaces of a cylinder. It consists of two circular bases and one rectangular lateral surface that wraps around. Formula: SA = 2πr² + 2πrh.

Circular Bases: The flat, round surfaces at the top and bottom of the cylinder, each with area πr².

Lateral Surface: The curved side surface that connects the two bases, with area 2πrh.

Complete methodology:
  1. Identify measurements: Determine radius (not diameter) and height
  2. Check units: Ensure consistent units for radius and height
  3. Apply formula: SA = 2πr² + 2πrh
  4. Calculate components: Find area of bases and lateral surface separately
  5. Sum components: Add all areas together
  6. Express answer: Include correct square units
Tip 1: The radius affects both base areas (r²) and lateral area (r), so it has significant impact.
Tip 2: Always convert diameter to radius before using the formula.

Tip 3: Distinguish between total surface area and lateral surface area based on context.

Tip 4: Remember that surface area is always expressed in square units.

Common errors: Forgetting to convert diameter to radius, using incorrect formula, confusing surface area with volume, forgetting to include both bases, mixing up radius and height, forgetting to square the units.
Exam preparation: Practice with various units, master dimensional analysis, understand the difference between total and lateral surface area, solve for missing values, compare different cylinders.
Surface area formulas and properties:

• Total Surface Area: SA = 2πr² + 2πrh

• Alternative Form: SA = 2πr(r + h)

• Lateral Surface Area: LSA = 2πrh

• Base Area: A = πr² (per base)

• Diameter Conversion: r = d/2

• Unit Relationship: Surface area is always in square units

• Component Breakdown: 2 circular bases + 1 rectangular lateral surface.

Exercise with Visualization: Surface Area Relationships
Exercise 6: Surface Area vs Radius/Height
Consider cylinders with varying dimensions:
Fixed height (h=10), varying radius: r=1, 2, 3, 4, 5
Fixed radius (r=3), varying height: h=2, 4, 6, 8, 10

Analysis: The chart shows how surface area changes with radius and height.

  • Surface area increases quadratically with radius (due to base areas)
  • Surface area increases linearly with height (due to lateral area)
  • Radius has a greater impact on surface area than height

Questions & Answers

Question: Why does the surface area formula have both r² and r terms? Shouldn't it be consistent?

Answer: The formula has both r² and r terms because surface area comes from different sources:

  • 2πr²: This represents the area of the two circular bases. Since area of a circle is πr², two bases contribute 2πr². This is where the r² term comes from.
  • 2πrh: This represents the lateral surface area. When you "unwrap" the curved surface, it forms a rectangle with width 2πr (the circumference) and height h. This is where the r term comes from.

So the r² term comes from the circular bases, while the r term comes from the lateral surface. Both are necessary to account for all surfaces of the cylinder!

Question: What's the difference between total surface area and lateral surface area? When do I use each?

Answer: The difference is:

  • Total Surface Area: SA = 2πr² + 2πrh (includes both circular bases AND the curved side)
  • Lateral Surface Area: LSA = 2πrh (includes only the curved side, excludes the bases)

Use total surface area when you need to cover the entire cylinder (like painting a closed can). Use lateral surface area when only the curved part matters (like a label on a can, or a hollow pipe).

Example: For a cylinder with r=3 cm and h=5 cm:

  • Total SA = 2π(9) + 2π(3)(5) = 18π + 30π = 48π ≈ 150.72 cm²
  • Lateral SA = 2π(3)(5) = 30π ≈ 94.2 cm²

Question: What's the relationship between surface area and volume of a cylinder?

Answer: Surface area and volume are related but measure different properties:

  • Volume: V = πr²h (measures space inside, cubic units)
  • Surface Area: SA = 2πr² + 2πrh (measures exterior surfaces, square units)

Key differences:

  • Volume depends on r²h, while surface area depends on r² + rh
  • Volume measures capacity, surface area measures coverage
  • Both increase when dimensions increase, but at different rates

For optimization problems, there's a relationship between volume and surface area that helps determine the most efficient dimensions!

Question: How do I handle problems where I'm given the circumference instead of the radius?

Answer: If you're given the circumference (C), you can find the radius using the circumference formula:

C = 2πr

Solving for r: r = C/(2π)

Example: If a cylinder has circumference 12.56 cm and height 8 cm:

  • Find radius: r = 12.56/(2×3.14) = 12.56/6.28 = 2 cm
  • Then use surface area formula: SA = 2πr² + 2πrh = 2π(4) + 2π(2)(8) = 8π + 32π = 40π ≈ 125.6 cm²

Always convert any given measurement to radius before using the surface area formula!

Question: How do I know which value to assign to radius and height when solving problems?

Answer: The key is to identify the dimensions correctly:

  • Radius: The distance from the center of the circular base to its edge. It's half the diameter.
  • Height: The perpendicular distance between the two circular bases. It's the "length" of the cylinder.

Tips for identification:

  • Look for the circular dimension (distance across the base) - that's diameter, so divide by 2 to get radius
  • Look for the straight-line distance along the side - that's the height
  • Radius is usually smaller than height in typical cylinders

Example: A soup can with "diameter 7 cm, height 10 cm" - radius is 3.5 cm, height is 10 cm.

The radius always goes into the formula as r, and height as h!