Surface Area of Cone

Question 1

Find the area of base and curved surface area of the cone given below.

Solution:

Given information:

• Vertical height of the cone (h) = 12 cm
• Slant height of the cone (l) = 13 cm
• Area of base of the cone = ?
• Curved surface area of the cone = ?
• Radius of base = r cm

Step 1: Find the radius of the base

In the figure, from right angled triangle POQ:

Using Pythagorean theorem: $PQ^2 = PO^2 + OQ^2$ ∴ $l^2 = h^2 + r^2$

Substituting the given values:

or, $13^2 = 12^2 + r^2$
or, $169 = 144 + r^2$
or, $169 - 144 = r^2$
or, $r^2 = 25$
so, $r = 5$ cm

∴ The radius of the base = 5 cm

Step 2: Calculate the area of base

We know that area of base = $\pi r^2$

Substituting the values:

$\text{Area of base} = \pi r^2 = \frac{22}{7} \times (5)^2$ $= \frac{22}{7} \times 25$ $= \frac{550}{7}$ $= 78.57$ cm²

Step 3: Calculate curved surface area

We know that curved surface area of cone = $\pi rl$

Substituting the values:

$\text{Curved surface area} = \pi rl = \frac{22}{7} \times 5 \times 13$ $= \frac{22 \times 65}{7}$ $= \frac{1430}{7}$ $= 204.28$ cm²

Final Answer:

Hence, the area of base = 78.57 cm² and curved surface area = 204.28 cm².

Verification:

• Pythagorean check: $l^2 = h^2 + r^2 = 12^2 + 5^2 = 144 + 25 = 169 = 13^2$ ✓
• Base area: $\pi r^2 = \frac{22}{7} \times 25 = 78.57 \text{ cm}^2$ ✓
• Curved surface area: $\pi rl = \frac{22}{7} \times 5 \times 13 = 204.28 \text{ cm}^2$ ✓
• Pythagorean triple: 5-12-13 triangle (extended 3-4-5 triple) ✓

Key formulas for cone:

• Pythagorean relationship: $l^2 = h^2 + r^2$ (slant height, vertical height, radius)
• Base area: $\text{Area} = \pi r^2$ (circular base)
• Curved surface area: $\text{CSA} = \pi rl$ (circumference × slant height)
• Total surface area: $\text{TSA} = \pi r^2 + \pi rl = \pi r(r + l)$

Geometric understanding:

• Right triangle formation: Vertical height, radius, and slant height form a right triangle
• 5-12-13 triangle: This is a scaled version of the 3-4-5 Pythagorean triple
• Curved surface: When unfolded, forms a sector of a circle with radius = slant height
• π approximation: Using $\frac{22}{7}$ for practical calculations

Note: This problem demonstrates the fundamental relationship between the three key measurements of a cone. The 5-12-13 right triangle is a common Pythagorean triple that appears frequently in geometry problems, making calculations and verification straightforward.

Question 2

Find the total surface area of a cone, if the diameter of base is 12 cm and the height is 8 cm.

Solution:

Given information:

• Diameter of the base of the cone (d) = 12 cm
• Radius of the base of the cone (r) = $\frac{d}{2} = \frac{12}{2} = 6$ cm
• Vertical height of the cone (h) = 8 cm
• Total surface area of the cone = ?

Step 1: Find the slant height (l)

From the figure, using Pythagorean theorem:

$l^2 = h^2 + r^2$

Substituting the given values:

$l^2 = 8^2 + 6^2$ $l^2 = 64 + 36$ $l^2 = 100$ $l = 10$ cm

∴ The slant height (l) = 10 cm

Step 2: Calculate the total surface area

We know that total surface area of cone = $\pi r(r + l)$

Substituting the values:

$\text{Total surface area} = \pi r(r + l)$ $= \frac{22}{7} \times 6 \times (6 + 10)$ $= \frac{22}{7} \times 6 \times 16$ $= \frac{22 \times 96}{7}$ $= \frac{2112}{7}$ $= 301.71$ cm²

Final Answer:

Hence, the total surface area of the cone is 301.71 cm².

Verification:

• Radius calculation: $r = \frac{d}{2} = \frac{12}{2} = 6 \text{ cm}$ ✓
• Pythagorean check: $l^2 = h^2 + r^2 = 8^2 + 6^2 = 64 + 36 = 100 = 10^2$ ✓
• Total surface area: $\pi r(r + l) = \frac{22}{7} \times 6 \times 16 = 301.71 \text{ cm}^2$ ✓
• Pythagorean triple: 6-8-10 triangle (scaled 3-4-5 triple) ✓

Alternative calculation breakdown:

• Base area: $\pi r^2 = \frac{22}{7} \times 6^2 = \frac{22 \times 36}{7} = 113.14 \text{ cm}^2$
• Curved surface area: $\pi rl = \frac{22}{7} \times 6 \times 10 = 188.57 \text{ cm}^2$
• Total surface area: $113.14 + 188.57 = 301.71 \text{ cm}^2$ ✓

Key formulas used:

• Radius from diameter: $r = \frac{d}{2}$
• Pythagorean relationship: $l^2 = h^2 + r^2$ (slant height formula)
• Total surface area: $\text{TSA} = \pi r^2 + \pi rl = \pi r(r + l)$
• Alternative form: $\text{TSA} = \text{Base area} + \text{Curved surface area}$

Problem-solving approach:

• Step 1: Convert diameter to radius for calculations
• Step 2: Use Pythagorean theorem to find slant height from height and radius
• Step 3: Apply total surface area formula using radius and slant height

Note: This problem showcases another common Pythagorean triple (6-8-10, which is 2 times the 3-4-5 triple). The total surface area formula πr(r + l) is a compact way to calculate both base area and curved surface area together. Always remember to convert diameter to radius when working with circular measurements.

Question 3

The sum of radius of base and slant height of a cone is 64 cm. If its total surface area is 2816 cm², find its curved surface area.

Solution:

Given information:

• Total surface area (TSA) = 2816 cm²
• Radius of base (r) + slant height (l) = 64 cm
• ∴ r + l = 64 ... (equation i)
• Curved surface area of the cone (CSA) = ?

Step 1: Find the radius using total surface area

We know that total surface area of a cone = $\pi r(r + l)$

Substituting the given values:

$2816 = \frac{22}{7} \times r \times 64$ $2816 \times 7 = r \times 22 \times 64$ $r = \frac{2816 \times 7}{22 \times 64}$ $r = \frac{19712}{1408}$ $r = 14$ cm

∴ The radius of the base = 14 cm

Step 2: Find the slant height

Putting the value of r in equation (i):

$14 + l = 64$ $l = 64 - 14 = 50$ cm

∴ The slant height = 50 cm

Step 3: Calculate the curved surface area

We know that curved surface area of cone = $\pi rl$

Substituting the values:

$\text{Curved surface area} = \pi rl = \frac{22}{7} \times 14 \times 50$ $= \frac{22 \times 14 \times 50}{7}$ $= \frac{15400}{7}$ $= 2200$ cm²

Final Answer:

Hence, the curved surface area of the cone is 2200 cm².

Verification:

• Sum constraint check: $r + l = 14 + 50 = 64 \text{ cm}$ ✓
• Total surface area check: $\pi r(r + l) = \frac{22}{7} \times 14 \times 64 = 2816 \text{ cm}^2$ ✓
• Curved surface area: $\pi rl = \frac{22}{7} \times 14 \times 50 = 2200 \text{ cm}^2$ ✓
• Base area: $\pi r^2 = \frac{22}{7} \times 14^2 = 616 \text{ cm}^2$
• TSA breakdown: $616 + 2200 = 2816 \text{ cm}^2$ ✓

Problem-solving approach:

• Step 1: Use total surface area formula with the constraint r + l = 64 to find radius
• Step 2: Apply the constraint equation to find slant height from radius
• Step 3: Calculate curved surface area using πrl formula

Key relationships used:

• Constraint equation: $r + l = 64$ (given condition)
• Total surface area: $\text{TSA} = \pi r(r + l)$ (substituting constraint)
• Curved surface area: $\text{CSA} = \pi rl$ (lateral surface formula)
• Alternative verification: $\text{TSA} = \text{Base area} + \text{CSA}$

Mathematical insight:

• Constraint substitution: Using r + l = 64 directly in TSA formula simplifies calculation
• Linear relationship: Once radius is found, slant height follows immediately
• Large numbers: This problem involves larger measurements, showing scalability of formulas
• Reverse engineering: Working backwards from surface area to find individual measurements

Note: This is a classic reverse problem where constraints are used to simplify calculations. The key insight is substituting the constraint r + l = 64 directly into the total surface area formula, which eliminates one variable and allows direct solution for the radius. The large numbers (2816, 2200) demonstrate that cone formulas work effectively at any scale.

Question 4

Find the length of circumference of base of a cone, if its total surface area and curved surface area are 1320 cm² and 704 cm².

Solution:

Given information:

• Total surface area (TSA) = 1320 cm²
• Curved surface area (CSA) = 704 cm²
• Length of circumference of base (C) = ?

Step 1: Find the base area

We know that total surface area of a cone = Area of base + curved surface area

Substituting the given values:

$1320 = \pi r^2 + 704$ $1320 - 704 = \pi r^2$ $616 = \frac{22}{7} r^2$

Step 2: Find the radius

Solving for r²:

$\frac{616 \times 7}{22} = r^2$ $\frac{4312}{22} = r^2$ $r^2 = 196$ $r = 14$ cm

∴ The radius of the base = 14 cm

Step 3: Calculate the circumference

We know that length of circumference (C) = $2\pi r$

Substituting the values:

$C = 2\pi r = 2 \times \frac{22}{7} \times 14$ $= \frac{2 \times 22 \times 14}{7}$ $= \frac{616}{7}$ $= 88$ cm

Final Answer:

Therefore, the length of circumference of the base of the cone is 88 cm.

Verification:

• Base area check: $\pi r^2 = \frac{22}{7} \times 14^2 = \frac{22 \times 196}{7} = 616 \text{ cm}^2$ ✓
• Total surface area check: $616 + 704 = 1320 \text{ cm}^2$ ✓
• Circumference calculation: $2\pi r = 2 \times \frac{22}{7} \times 14 = 88 \text{ cm}$ ✓
• Radius verification: $r^2 = 196$ ∴ $r = 14$ cm ✓

Problem-solving approach:

• Step 1: Use the relationship TSA = Base area + CSA to find base area
• Step 2: Apply the circle area formula to find radius from base area
• Step 3: Calculate circumference using the circle circumference formula

Key formulas used:

• Total surface area: $\text{TSA} = \pi r^2 + \text{CSA}$ (base area + curved surface area)
• Base area: $\text{Area} = \pi r^2$ (circular base)
• Circumference: $C = 2\pi r$ (perimeter of circular base)
• Relationship: $\text{Base area} = \text{TSA} - \text{CSA}$

Mathematical insight:

• Direct subtraction: TSA - CSA immediately gives base area
• Perfect square: r² = 196 = 14², making radius calculation simple
• Clean numbers: All calculations result in whole numbers
• Verification path: Multiple ways to check the answer ensure accuracy

Note: This problem demonstrates how knowing two of the three main surface area components (total, base, curved) allows us to find the third. The clean arithmetic (r² = 196 = 14²) makes this an excellent example for understanding the relationships between cone measurements. The circumference connects the radius to the linear measurement around the base.