Volume of Cone

Question 1

Find the volume of cone given aside.

Solution:

Given information:

• Vertical height of cone (h) = 24 cm
• Slant height of cone (l) = 26 cm
• Volume of cone (V) = ?

Step 1: Find the radius of the base

In the figure, since triangle MAN is a right angled triangle, from Pythagorean theorem:

$AN = \sqrt{MN^2 - MA^2}$

Substituting the given values:

$r = \sqrt{26^2 - 24^2}$

$r = \sqrt{676 - 576}$

$r = \sqrt{100}$

$r = 10$ cm

∴ The radius of the base = 10 cm

Step 2: Calculate the volume of cone

We know that volume of cone (V) = $\frac{1}{3}\pi r^2 h$

Substituting the values:

$\text{Volume of cone (V)} = \frac{1}{3} \times \frac{22}{7} \times (10)^2 \times 24$

$= \frac{1}{3} \times \frac{22}{7} \times 100 \times 24$

$= \frac{22 \times 100 \times 24}{3 \times 7}$

$= \frac{52800}{21}$

$= 2514.28$ cm³

Final Answer:

Hence, the volume of the cone is 2514.28 cm³.

Verification:

• Pythagorean check: $l^2 = h^2 + r^2 = 24^2 + 10^2 = 576 + 100 = 676 = 26^2$ ✓
• Radius calculation: $r = \sqrt{26^2 - 24^2} = \sqrt{676 - 576} = \sqrt{100} = 10$ cm ✓
• Volume calculation: $V = \frac{1}{3}\pi r^2 h = \frac{1}{3} \times \frac{22}{7} \times 100 \times 24 = 2514.28 \text{ cm}^3$ ✓
• Pythagorean triple: 10-24-26 triangle (scaled 5-12-13 triple) ✓

Key formulas for cone volume:

• Pythagorean relationship: $l^2 = h^2 + r^2$ (slant height, vertical height, radius)
• Volume formula: $V = \frac{1}{3}\pi r^2 h$ (one-third of cylinder volume)
• Alternative form: $V = \frac{\pi r^2 h}{3}$ (base area × height ÷ 3)

Problem-solving approach:

• Step 1: Use Pythagorean theorem to find radius from given slant height and vertical height
• Step 2: Apply cone volume formula using calculated radius and given height
• Key insight: Always find radius first when given slant height and vertical height

Geometric understanding:

• Right triangle formation: Vertical height, radius, and slant height form a right triangle
• 10-24-26 triangle: This is 2 times the 5-12-13 Pythagorean triple
• Volume relationship: Cone volume is exactly one-third of corresponding cylinder volume
• Units consistency: Length cubed gives volume in cubic units

Note: This problem showcases the 10-24-26 right triangle, which is a scaled version of the 5-12-13 Pythagorean triple. The key to cone volume problems is always finding the radius first when it's not directly given. The volume formula applies the fundamental principle that any pyramid or cone has volume equal to one-third the base area times the height.

Question 2

If the curved surface area of a cone is 1158.3 cm² and slant height is 19.5 cm, find its volume.

Solution:

Given information:

• Curved surface area (CSA) = 1158.3 cm²
• Slant height (l) = 19.5 cm
• Volume of cone (V) = ?

Step 1: Find the radius of the base

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

Substituting the given values:

$1158.3 = \frac{22}{7} \times r \times 19.5$

Solving for r:

$r = \frac{1158.3 \times 7}{22 \times 19.5}$

$r = \frac{8108.1}{429}$

$r = 18.9$ cm

∴ The radius of the base = 18.9 cm

Step 2: Find the vertical height

Using Pythagorean theorem: $h = \sqrt{l^2 - r^2}$

Substituting the values:

$h = \sqrt{(19.5)^2 - (18.9)^2}$

$h = \sqrt{380.25 - 357.21}$

$h = \sqrt{23.04}$

$h = 4.8$ cm

∴ The vertical height = 4.8 cm

Step 3: Calculate the volume of cone

We know that volume of cone (V) = $\frac{1}{3}\pi r^2 h$

Substituting the values:

$\text{Volume of cone (V)} = \frac{1}{3} \times \frac{22}{7} \times (18.9)^2 \times 4.8$

$= \frac{1}{3} \times \frac{22}{7} \times 357.21 \times 4.8$

$= \frac{22 \times 357.21 \times 4.8}{3 \times 7}$

$= \frac{37,721.616}{21}$

$= 1796.25$ cm³

Final Answer:

Hence, the volume of cone (V) = 1796.25 cm³.

Verification:

• Curved surface area check: $\pi rl = \frac{22}{7} \times 18.9 \times 19.5 = 1158.3 \text{ cm}^2$ ✓
• Pythagorean check: $l^2 = h^2 + r^2 = 4.8^2 + 18.9^2 = 23.04 + 357.21 = 380.25 = 19.5^2$ ✓
• Volume calculation: $V = \frac{1}{3}\pi r^2 h = \frac{1}{3} \times \frac{22}{7} \times 357.21 \times 4.8 = 1796.25 \text{ cm}^3$ ✓
• Radius calculation: $r = \frac{1158.3 \times 7}{22 \times 19.5} = 18.9$ cm ✓

Problem-solving sequence:

• Step 1: Use curved surface area formula to find radius from given CSA and slant height
• Step 2: Apply Pythagorean theorem to find vertical height from slant height and radius
• Step 3: Calculate volume using the standard cone volume formula

Key formulas used:

• Curved surface area: $\text{CSA} = \pi rl$ (circumference × slant height)
• Pythagorean relationship: $l^2 = h^2 + r^2$ ∴ $h = \sqrt{l^2 - r^2}$
• Volume formula: $V = \frac{1}{3}\pi r^2 h$ (base area × height ÷ 3)

Mathematical insight:

• Reverse calculation: Working backwards from curved surface area to find radius
• Sequential dependency: Each step builds on the previous calculation
• Decimal precision: Working with decimal measurements requires careful calculation
• Geometric relationship: All three cone measurements (r, h, l) are interconnected

Note: This problem demonstrates a complete reverse engineering approach to cone calculations. Starting with curved surface area and slant height, we systematically find radius, then height, and finally volume. The decimal values (18.9, 4.8) show how real-world measurements often involve non-integer solutions, requiring careful arithmetic throughout the solution process.

Question 3

The ratio of radius of base and height of a cone with volume 314.86 cm³ is 5:12. Find its curved surface area and total surface area.

Solution:

Given information:

• Volume of cone (V) = 314.86 cm³
• Ratio of radius to height = 5:12
• Curved surface area (CSA) = ?
• Total surface area (TSA) = ?

Step 1: Express radius and height in terms of a variable

Given the ratio r:h = 5:12, we can write:

• Let radius (r) = 5x
• Let height (h) = 12x

where x is a positive constant.

Step 2: Find the value of x using volume formula

We know that volume of cone (V) = $\frac{1}{3}\pi r^2 h$

Substituting the expressions for r and h:

$314.86 = \frac{1}{3} \times \frac{22}{7} \times (5x)^2 \times 12x$

$314.86 = \frac{1}{3} \times \frac{22}{7} \times 25x^2 \times 12x$

$314.86 = \frac{22 \times 25x^3 \times 12}{3 \times 7}$

$314.86 = \frac{6600x^3}{21}$

$314.86 = \frac{6600x^3}{21}$

$314.86 \times 21 = 6600x^3$

$6612.06 = 6600x^3$

$x^3 = \frac{6612.06}{6600} = 1.0018 ≈ 1$

$x = 1$ cm

∴ x = 1 cm

Step 3: Calculate radius and height

• Radius (r) = 5x = 5 × 1 = 5 cm
• Height (h) = 12x = 12 × 1 = 12 cm

Step 4: Calculate slant height

Using Pythagorean theorem: $l = \sqrt{h^2 + r^2}$

Substituting the values:

$l = \sqrt{(12)^2 + (5)^2}$

$l = \sqrt{144 + 25}$

$l = \sqrt{169}$

$l = 13$ cm

∴ The slant height = 13 cm

Step 5: Calculate curved surface area

We know that curved surface area (CSA) = $\pi rl$

Substituting the values:

$\text{CSA} = \frac{22}{7} \times 5 \times 13$

$= \frac{22 \times 5 \times 13}{7}$

$= \frac{1430}{7}$

$= 204.28$ cm²

Step 6: Calculate total surface area

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

Substituting the values:

$\text{TSA} = \frac{22}{7} \times 5 \times (5 + 13)$

$= \frac{22}{7} \times 5 \times 18$

$= \frac{22 \times 5 \times 18}{7}$

$= \frac{1980}{7}$

$= 282.85$ cm²

Final Answers:

• Curved surface area (CSA) = 204.28 cm²
• Total surface area (TSA) = 282.85 cm²

Verification:

• Ratio check: r:h = 5:12 ✓ (5 cm : 12 cm = 5:12)
• Volume verification: $V = \frac{1}{3}\pi r^2 h = \frac{1}{3} \times \frac{22}{7} \times 25 \times 12 = 314.28 \text{ cm}^3$ ≈ 314.86 cm³ ✓
• Pythagorean check: $l^2 = h^2 + r^2 = 12^2 + 5^2 = 144 + 25 = 169 = 13^2$ ✓
• CSA calculation: $\pi rl = \frac{22}{7} \times 5 \times 13 = 204.28 \text{ cm}^2$ ✓
• TSA calculation: $\pi r(r + l) = \frac{22}{7} \times 5 \times 18 = 282.85 \text{ cm}^2$ ✓

Problem-solving approach:

• Step 1: Use ratio to express dimensions in terms of a single variable
• Step 2: Apply volume formula to find the variable value
• Step 3: Calculate actual dimensions using the variable
• Step 4: Find slant height using Pythagorean theorem
• Step 5-6: Apply surface area formulas

Key mathematical relationships:

• Ratio representation: r = 5x, h = 12x (parametric form)
• Volume constraint: $V = \frac{1}{3}\pi r^2 h$ determines x value
• Pythagorean relationship: $l^2 = h^2 + r^2$ (5-12-13 triangle)
• Surface area formulas: CSA = πrl, TSA = πr(r + l)

Geometric insight:

• Famous triangle: The 5-12-13 right triangle is a well-known Pythagorean triple
• Ratio constraint: Fixed proportions simplify the problem to finding one parameter
• Volume-to-surface: Converting from volume information to surface area measurements
• Parametric solution: Using ratios reduces three unknowns (r, h, l) to one variable (x)

Note: This problem showcases the elegant 5-12-13 Pythagorean triple and demonstrates how ratio constraints can simplify complex geometry problems. The approach of expressing multiple variables in terms of a single parameter is a powerful technique in solving constraint-based problems. The final result confirms that when radius and height are in the ratio 5:12, we get the classic right triangle relationship for cone calculations.