Solved Exercises on Surface Area of Cones in Grade 8

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

Solution: Exercises 1 to 3
1 Basic Cone Surface Area
Exercise 1
Find the surface area of a cone with radius 6 cm and slant height 10 cm. Use π ≈ 3.14.
Definition:

Surface Area of Cone: The total area of all surfaces of a cone. The formula is SA = πr² + πrl, where r is the radius of the circular base and l is the slant height. This includes the circular base and the curved lateral surface.

Surface Area Calculation Method:
  1. Identify the radius (r) and slant height (l)
  2. Apply the formula SA = πr² + πrl
  3. Calculate the area of the circular base: πr²
  4. Calculate the lateral surface area: πrl
  5. Add the two areas together
  6. Include the correct units (square units)
Given Values
r = 6 cm, l = 10 cm
Formula
SA = πr² + πrl
Result
SA = 301.44 cm²
Step 1: Write down the formula

SA = πr² + πrl

Step 2: Calculate area of the circular base

πr² = π × (6)² = π × 36 = 36π

Using π ≈ 3.14: 36π ≈ 36 × 3.14 = 113.04 cm²

Step 3: Calculate the lateral surface area

πrl = π × 6 × 10 = 60π

Using π ≈ 3.14: 60π ≈ 60 × 3.14 = 188.4 cm²

Step 4: Add both areas

SA = 113.04 + 188.4 = 301.44 cm²

SA = 301.44 cm²
Final answer:

The surface area of the cone is 301.44 square centimeters.

Applied rules:

• Surface Area Formula: SA = πr² + πrl

• Components: Includes circular base and lateral surface

• Units: Surface area is measured in square units

2 Diameter to Radius Conversion
Exercise 2
A conical party hat has a diameter of 12 inches and a slant height of 8 inches. 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 = 12 in, l = 8 in
Convert to Radius
r = 6 in
Surface Area
SA = 263.76 in²
Step 1: Convert diameter to radius

r = d/2 = 12/2 = 6 inches

Step 2: Apply the surface area formula

SA = πr² + πrl = π(6)² + π(6)(8)

Step 3: Calculate each component

Area of base: πr² = π × 36 = 36π ≈ 36 × 3.14 = 113.04 in²

Lateral surface: πrl = π × 6 × 8 = 48π ≈ 48 × 3.14 = 150.72 in²

Step 4: Add both areas

SA = 113.04 + 150.72 = 263.76 in²

SA = 263.76 in²
Final answer:

The surface area of the party hat is 263.76 square inches.

Applied rules:

• Diameter to Radius: r = d/2

• Surface Area Formula: SA = πr² + πrl

• Unit Consistency: Keep all measurements in the same units

3 Lateral Surface Area Only
Exercise 3
Find the lateral surface area of a cone with radius 5 inches and slant height 13 inches. Use π ≈ 3.14.
Definition:

Lateral Surface Area: The area of just the curved side of the cone, excluding the circular base. The formula is LSA = πrl.

Given Values
r = 5 in, l = 13 in
Formula
LSA = πrl
Lateral Surface Area
LSA = 204.1 in²
Step 1: Write down the lateral surface area formula

LSA = πrl

Step 2: Substitute the known values

LSA = π × 5 × 13

Step 3: Calculate

LSA = 3.14 × 5 × 13

LSA = 3.14 × 65

LSA = 204.1 in²

LSA = 204.1 in²
Final answer:

The lateral surface area of the cone is 204.1 square inches.

Applied rules:

• Lateral Surface Area: LSA = πrl (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,...
\(SA = \pi r^2 + \pi rl\)
Surface Area of Cone
Total Surface Area
SA = πr² + πrl
Includes base and lateral surface
Lateral Surface Area
LSA = πrl
Only curved side
Base Area
A = πr²
Area of circular base
Diameter to Radius
r = d/2
Always convert first
Alternative Formula
SA = πr(r + l)
Factored form
Components: 1 circular base + 1 sector (lateral surface when unwrapped)
Pythagorean Relationship: l² = r² + h² (slant height, radius, height)
Key definitions:

Cone: A three-dimensional shape with a circular base that tapers smoothly to a point called the apex or vertex.

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.

Slant Height (l): The distance from the apex to the edge of the base along the surface.

Height (h): The perpendicular distance from the base to the apex.

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 = πr² + πrl
  4. Calculate components: Find area of base 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 conical roof has a diameter of 14 feet and a slant height of 10 feet. If roofing material costs $5 per square foot, how much would it cost to cover the entire roof? Use π ≈ 3.14.
Definition:

Real-World Application: Calculating surface area to determine material costs for covering conical structures like roofs, tents, or party hats.

Given Values
d = 14 ft, l = 10 ft
Convert to Radius
r = 7 ft
Cost
$659.40
Step 1: Find the radius

r = d/2 = 14/2 = 7 feet

Step 2: Calculate the total surface area

SA = πr² + πrl = π(7)² + π(7)(10)

SA = π(49) + π(70) = 49π + 70π = 119π

SA = 119 × 3.14 = 373.66 ft²

Step 3: Calculate the total cost

Cost = Surface Area × Price per square foot

Cost = 373.66 × $5 = $1,868.30

$1,868.30
Final answer:

The cost to cover the entire roof would be $1,868.30.

Applied rules:

• Total Surface Area: SA = πr² + πrl for entire surface

• Diameter to Radius: r = d/2

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

5 Finding Missing Dimensions
Exercise 5
The surface area of a cone is 188.4 square inches and the radius is 3 inches. Find the slant height of the cone. Use π ≈ 3.14.
Definition:

Algebraic Manipulation: When solving for missing dimensions, rearrange the surface area formula algebraically. From SA = πr² + πrl, we get l = (SA - πr²)/(πr).

Given Values
SA = 188.4 in², r = 3 in
Formula Rearranged
l = (SA - πr²)/(πr)
Slant Height
l = 17 in
Step 1: Start with the surface area formula

SA = πr² + πrl

Step 2: Solve for l

188.4 = π(3)² + π(3)l

188.4 = π(9) + 3πl

188.4 = 9π + 3πl

188.4 = 9(3.14) + 3(3.14)l

188.4 = 28.26 + 9.42l

Step 3: Isolate l

188.4 - 28.26 = 9.42l

160.14 = 9.42l

l = 160.14/9.42 = 17 inches

Step 4: Verify the solution

SA = π(3)² + π(3)(17) = 9π + 51π = 60π = 60 × 3.14 = 188.4 ✓

l = 17 inches
Final answer:

The slant height of the cone is 17 inches.

Applied rules:

• Algebraic Rearrangement: Isolate the unknown variable

• Solving Equations: Use inverse operations to isolate l

• Verification: Check by substituting back into original formula

Surface Area of Cones Laws, Methods, and Properties
\(SA = \pi r^2 + \pi rl\)
Surface Area of Cone
Key definitions:

Surface Area of Cone: The measure of the total area of all surfaces of a cone. It consists of one circular base and one curved lateral surface. Formula: SA = πr² + πrl.

Circular Base: The flat, round surface at the bottom of the cone, with area πr².

Lateral Surface: The curved side surface that connects the base to the apex, with area πrl.

Complete methodology:
  1. Identify measurements: Determine radius and slant height
  2. Check units: Ensure consistent units
  3. Apply formula: SA = πr² + πrl
  4. Calculate components: Find area of base 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 area (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, confusing slant height with vertical height, using incorrect formula, confusing surface area with volume, forgetting to include the base, mixing up radius and slant height.
Exam preparation: Practice with various units, master dimensional analysis, understand the difference between total and lateral surface area, solve for missing values, compare different cones.
Surface area formulas and properties:

• Total Surface Area: SA = πr² + πrl

• Alternative Form: SA = πr(r + l)

• Lateral Surface Area: LSA = πrl

• Base Area: A = πr²

• Diameter Conversion: r = d/2

• Pythagorean Relationship: l² = r² + h² (relates slant height, radius, and height)

• Unit Relationship: Surface area is always in square units

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

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

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

  • πr²: This represents the area of the circular base. Since area of a circle is πr², this is where the r² term comes from.
  • πrl: This represents the lateral surface area. When you "unwrap" the curved surface, it forms a sector of a circle with radius l and arc length 2πr. The area of this sector is πrl. This is where the r term comes from.

So the r² term comes from the base, while the r term in πrl comes from the lateral surface. Both are necessary to account for all surfaces of the cone!

Question: What's the difference between slant height and vertical height in a cone? How do I know which one to use?

Answer: The differences are:

  • Vertical Height (h): The perpendicular distance from the base to the apex. Used in volume calculations.
  • Slant Height (l): The distance from the apex to the edge of the base along the surface. Used in surface area calculations.

Use slant height for surface area and vertical height for volume. They're related by the Pythagorean theorem: l² = r² + h².

For surface area: SA = πr² + πrl (uses slant height)

For volume: V = (1/3)πr²h (uses vertical height)

If you're given vertical height and need slant height, use l = √(r² + h²).

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

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

  • Volume: V = (1/3)πr²h (measures space inside, cubic units)
  • Surface Area: SA = πr² + πrl (measures exterior surfaces, square units)

Key differences:

  • Volume depends on r²h, while surface area depends on r² + rl
  • 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 cone has circumference 18.84 cm and slant height 8 cm:

  • Find radius: r = 18.84/(2×3.14) = 18.84/6.28 = 3 cm
  • Then use surface area formula: SA = πr² + πrl = π(9) + π(3)(8) = 9π + 24π = 33π ≈ 103.62 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 slant 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.
  • Slant Height: The distance from the apex to the edge of the base along the surface. It's longer than the vertical height.
  • Vertical Height: The perpendicular distance from the base to the apex (not used in surface area).

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 diagonal distance from tip to edge - that's the slant height
  • Slant height is typically longer than radius

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