Solved Exercises on Surface Area of Spheres in Grade 8

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

Solution: Exercises 1 to 3
1 Basic Sphere Surface Area
Exercise 1
Find the surface area of a sphere with radius 7 cm. Use π ≈ 3.14.
Definition:

Surface Area of Sphere: The total area of the curved surface of a sphere. The formula is SA = 4πr², where r is the radius of the sphere. A sphere has only one continuous surface.

Surface Area Calculation Method:
  1. Identify the radius (r)
  2. Apply the formula SA = 4πr²
  3. Substitute the value and calculate
  4. Include the correct units (square units)
Given Value
r = 7 cm
Formula
SA = 4πr²
Result
SA = 615.44 cm²
Step 1: Write down the formula

SA = 4πr²

Step 2: Substitute the known value

SA = 4 × π × (7)²

SA = 4 × π × 49

Step 3: Calculate

SA = 4 × 3.14 × 49

SA = 12.56 × 49

SA = 615.44 cm²

SA = 615.44 cm²
Final answer:

The surface area of the sphere is 615.44 square centimeters.

Applied rules:

• Surface Area Formula: SA = 4πr² for spheres

• Units: Surface area is measured in square units

• Single Surface: Spheres have only one continuous surface

2 Diameter to Radius Conversion
Exercise 2
A basketball has a diameter of 24 cm. 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 Value
d = 24 cm
Convert to Radius
r = 12 cm
Surface Area
SA = 1,808.64 cm²
Step 1: Convert diameter to radius

r = d/2 = 24/2 = 12 cm

Step 2: Apply the surface area formula

SA = 4πr² = 4 × π × (12)²

Step 3: Calculate

SA = 4 × 3.14 × 144

SA = 12.56 × 144

SA = 1,808.64 cm²

SA = 1,808.64 cm²
Final answer:

The surface area of the basketball is 1,808.64 square centimeters.

Applied rules:

• Diameter to Radius: r = d/2

• Surface Area Formula: SA = 4πr²

• Unit Consistency: Keep all measurements in the same units

3 Finding Missing Dimensions
Exercise 3
The surface area of a sphere is 1,256 square inches. Find the radius of the sphere. Use π ≈ 3.14.
Definition:

Algebraic Manipulation: When solving for missing dimensions, rearrange the surface area formula algebraically. From SA = 4πr², we get r² = SA/(4π) and r = √[SA/(4π)].

Given Value
SA = 1,256 in²
Formula Rearranged
r² = SA/(4π)
Radius
r = 10 in
Step 1: Start with the surface area formula

SA = 4πr²

Step 2: Solve for r²

1,256 = 4 × π × r²

1,256 = 4 × 3.14 × r²

1,256 = 12.56 × r²

r² = 1,256/12.56

r² = 100

Step 3: Solve for r

r = √100 = 10 inches

Step 4: Verify the solution

SA = 4πr² = 4 × 3.14 × 100 = 1,256 ✓

r = 10 inches
Final answer:

The radius of the sphere is 10 inches.

Applied rules:

• Algebraic Rearrangement: Isolate the unknown variable

• Square Root: Take the positive root since radius is positive

• Verification: Check by substituting back into original formula

Rules and methods, laws,...
\(SA = 4\pi r^2\)
Surface Area of Sphere
Surface Area Formula
SA = 4πr²
Depends on radius squared
Diameter to Radius
r = d/2
Always convert first
Volume
V = (4/3)πr³
Related property
Finding Radius
r = √(SA/4π)
When surface area known
Finding Diameter
d = 2√(SA/4π)
When surface area known
Single Surface: Spheres have only one continuous surface
Maximum Efficiency: Spheres have maximum volume for minimum surface area
Key definitions:

Sphere: A three-dimensional shape where all points on the surface are equidistant from the center.

Radius (r): The distance from the center of the sphere to any point on its surface.

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

Center: The point inside the sphere that is equidistant from all points on the surface.

Surface Area: The total area of the curved surface of the sphere, 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 = 4πr²
  4. Perform calculations: Square the radius first, then multiply by 4π
  5. 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: The radius squared relationship means small changes in radius cause large changes in surface area.

Solution: Exercises 4 to 5
4 Real-World Application
Exercise 4
A spherical water balloon has a diameter of 10 inches. If the rubber material costs $0.05 per square inch, how much would it cost to make the balloon? Use π ≈ 3.14.
Definition:

Real-World Application: Calculating surface area to determine material costs for covering spherical objects like balloons, balls, or globes.

Given Value
d = 10 in
Convert to Radius
r = 5 in
Cost
$15.70
Step 1: Find the radius

r = d/2 = 10/2 = 5 inches

Step 2: Calculate the surface area

SA = 4πr² = 4 × 3.14 × (5)²

SA = 4 × 3.14 × 25

SA = 314 in²

Step 3: Calculate the total cost

Cost = Surface Area × Price per square inch

Cost = 314 × $0.05 = $15.70

$15.70
Final answer:

The cost to make the balloon would be $15.70.

Applied rules:

• Surface Area Formula: SA = 4πr²

• Diameter to Radius: r = d/2

• Real-World Context: Cost = Surface Area × Unit Price

5 Comparative Analysis
Exercise 5
Sphere A has radius 3 cm. Sphere B has radius 6 cm. How many times greater is the surface area of Sphere B compared to Sphere A? Use π ≈ 3.14.
Definition:

Comparative Analysis: Calculating surface areas to compare their ratios. Since surface area depends on radius squared, doubling the radius increases surface area by a factor of 2² = 4.

Sphere A
SA_A = 113.04 cm²
Sphere B
SA_B = 452.16 cm²
Ratio
SA_B/SA_A = 4
Step 1: Calculate surface area of Sphere A

SA_A = 4πr² = 4 × 3.14 × (3)²

SA_A = 4 × 3.14 × 9 = 113.04 cm²

Step 2: Calculate surface area of Sphere B

SA_B = 4πr² = 4 × 3.14 × (6)²

SA_B = 4 × 3.14 × 36 = 452.16 cm²

Step 3: Compare surface areas

Ratio = SA_B/SA_A = 452.16/113.04 = 4

Step 4: Analyze the result

Since the radius of Sphere B is twice that of Sphere A (6 vs 3), and surface area is proportional to r², the surface area ratio is 2² = 4. This confirms that surface area scales with the square of the radius.

Sphere B's surface area is 4 times greater than Sphere A's
Final answer:

Sphere B's surface area is 4 times greater than Sphere A's surface area.

Applied rules:

• Surface Area Formula: SA = 4πr² for both spheres

• Comparative Analysis: Calculate each surface area separately

• Squaring Effect: When radius doubles, surface area increases by 2² = 4 times

Surface Area of Spheres Laws, Methods, and Properties
\(SA = 4\pi r^2\)
Surface Area of Sphere
Key definitions:

Surface Area of Sphere: The measure of the total area of the curved surface of a sphere, calculated as four times π times the radius squared. Formula: SA = 4πr².

Spherical Surface: The curved outer boundary of the sphere, with no edges or vertices.

Equidistant Property: Every point on the sphere's surface is the same distance from the center.

Complete methodology:
  1. Identify measurements: Determine radius (not diameter)
  2. Check units: Ensure consistent units
  3. Apply formula: SA = 4πr²
  4. Calculate: Square the radius first, then multiply by 4π
  5. Express answer: Include correct square units
Tip 1: The radius squared relationship means small changes in radius cause large changes in surface area.
Tip 2: Always convert diameter to radius before using the formula.

Tip 3: Remember that surface area is always expressed in square units.

Tip 4: Spheres have the minimum surface area for a given volume among all shapes.

Common errors: Forgetting to convert diameter to radius, using incorrect formula, confusing surface area with volume, forgetting to square the radius, forgetting to square the units.
Exam preparation: Practice with various units, master dimensional analysis, understand the squared relationship of radius to surface area, solve for missing values, compare with other shapes.
Surface area formulas and properties:

• Basic Formula: SA = 4πr²

• Diameter Conversion: r = d/2

• Finding Radius: r = √(SA/4π)

• Finding Diameter: d = 2√(SA/4π)

• Volume: V = (4/3)πr³

• Unit Relationship: Surface area is always in square units

• Squaring Effect: Surface area is proportional to r², so if radius doubles, surface area increases by 4 times.

Exercise with Visualization: Surface Area Relationships
Exercise 6: Surface Area vs Radius
Consider spheres with varying radii:
r = 1, 2, 3, 4, 5, 6 cm
Showing the quadratic relationship between radius and surface area

Analysis: The chart shows how surface area increases rapidly with radius.

  • Surface area increases quadratically with radius (SA ∝ r²)
  • Small increases in radius cause large increases in surface area
  • This demonstrates the squared relationship in the formula

Questions & Answers

Question: Why is the surface area formula 4πr²? Where does the 4 come from?

Answer: The 4 coefficient comes from calculus and the relationship between a sphere and its circumscribed cylinder. If you imagine a sphere inscribed in a cylinder where the cylinder's diameter equals the sphere's diameter and the cylinder's height equals the sphere's diameter, the sphere's surface area is exactly 2/3 of the cylinder's surface area.

The cylinder's surface area would be 2πr(2r) + 2πr² = 6πr², so the sphere's surface area is (2/3) × 6πr² = 4πr².

Alternatively, if you think of the sphere as being made up of infinitely many tiny pyramids with their apex at the center, the surface area would be the sum of all the base areas of these pyramids, which leads to the 4πr² formula.

Question: How does the surface area of a sphere compare to that of a cylinder or cone with the same radius?

Answer: Let's compare when radius is the same (r) and for the cylinder and cone, height equals diameter (2r):

  • Sphere: SA = 4πr²
  • Cylinder: SA = 2πr² + 2πrh = 2πr² + 2πr(2r) = 6πr²
  • Cone: SA = πr² + πrl (where l is slant height)

So for the same radius and comparable height, the sphere has less surface area than the cylinder but more than the circular base alone. The sphere is the most efficient shape for containing volume with minimum surface area.

This is why soap bubbles and water droplets form spheres - they minimize surface area for a given volume!

Question: What's the difference between the surface area and volume of a sphere? They seem related.

Answer: While both involve the sphere's radius, they measure different things:

  • Surface Area: Measures the area of the outer surface. Formula: SA = 4πr². Measured in square units.
  • Volume: Measures the space inside the sphere. Formula: V = (4/3)πr³. Measured in cubic units.

Think of it this way:

  • Surface Area = how much wrapping paper covers the sphere
  • Volume = how much liquid fills the sphere

Notice that surface area uses r² while volume uses r³. Both depend on π, but for different geometric properties!

Question: Why does the radius squared relationship make such a big difference in surface area?

Answer: The squared relationship means that surface area changes by the square of the factor by which the radius changes:

  • If radius doubles (×2), surface area increases by 2² = 4 times
  • If radius triples (×3), surface area increases by 3² = 9 times
  • If radius increases by 50% (×1.5), surface area increases by 1.5² = 2.25 times

This is why a small increase in radius causes a disproportionately large increase in surface area. For example, going from radius 3 to 4 (a 33% increase) results in surface area changing from 4π(9) to 4π(16), which is nearly 78% increase!

This squared relationship is unique to two-dimensional properties and explains why spheres pack efficiently in nature.

Question: How do I handle problems with partial spheres or hemispheres?

Answer: For hemispheres and partial spheres:

  • Hemisphere: Half of a sphere plus the circular base, so SA = (1/2) × 4πr² + πr² = 2πr² + πr² = 3πr²
  • Quarter sphere: (1/4) × 4πr² plus the appropriate base areas
  • Any fraction: Multiply the full sphere surface area by the fraction

Example: For a hemisphere with radius 5 cm:

  • Curved surface: (1/2) × 4π(5)² = 2π(25) = 50π cm²
  • Base area: π(5)² = 25π cm²
  • Total SA: 50π + 25π = 75π ≈ 235.5 cm²

Don't forget to include the base area when the flat surface is part of the surface area!