CBSE Class 9 Maths: Surface Area and Volume - Exercise 13.7 (Cones)

Understanding Right Circular Cones and Their Volume

A right circular cone is a three-dimensional geometric shape that tapers smoothly from a flat circular base to a point called the apex or vertex. The "right circular" part means that the line segment joining the vertex to the center of the base is perpendicular to the base. This perpendicular distance is known as the height (h) of the cone. The radius of the circular base is denoted by r, and the distance from the apex to any point on the circumference of the base is called the slant height (l).

These three dimensions – radius (r), height (h), and slant height (l) – are intimately related. If you imagine cutting a cone from its apex to its base along its slant height and flattening it, you'd see a sector of a circle and its circular base. The relationship between r, h, and l forms a right-angled triangle, with the slant height l as the hypotenuse, and the height h and radius r as the other two sides. Therefore, by the Pythagorean theorem, we have:

l² = r² + h²

This formula is absolutely critical, as you'll often need to find one of these dimensions when the other two are given.

Now, let's focus on the primary concept of Exercise 13.7: Volume of a Cone.

The volume of a cone is one-third the product of the area of its base and its height. Since the base is a circle, its area is πr².

Volume of a Cone (V) = (1/3) × (Area of Base) × Height
V = (1/3)πr²h

It's important to remember this formula and understand its components. Always ensure your units for radius and height are consistent (e.g., both in cm or both in m) before performing calculations. For π, unless specified otherwise, use 22/7 or 3.14.

Key Definitions for Cones

Right Circular Cone

A three-dimensional solid having a circular base and a single vertex, with the line segment from the vertex to the center of the base being perpendicular to the base.

Radius (r)

The radius of the circular base of the cone.

Height (h)

The perpendicular distance from the apex (vertex) to the center of the circular base.

Slant Height (l)

The distance from the apex to any point on the circumference of the base. It can be found using the formula: l = √(r² + h²) .

Volume of a Cone (V)

The amount of space occupied by the cone, calculated as V = (1/3)πr²h.

Worked Examples: Calculating Volume of Cones

• Example 1: A conical pit of top diameter 3.5 m is 12 m deep. What is its capacity in kilolitres? A: Step 1: Identify given values and goal. Given: Diameter = 3.5 m, Depth (height) h = 12 m. Goal: Find capacity (volume) in kilolitres. Step 2: Calculate radius (r). Radius r = Diameter / 2 = 3.5 m / 2 = 1.75 m. Step 3: Apply the volume formula. Volume of cone V = (1/3)πr²h V = (1/3) × (22/7) × (1.75)² × 12 V = (1/3) × (22/7) × (1.75 × 1.75) × 12 V = (1/3) × (22/7) × (3.0625) × 12 V = (1/3) × 22 × (0.25 × 1.75) × 12 (Since 1.75/7 = 0.25) V = 22 × (0.25 × 1.75) × 4 (Cancel 3 with 12) V = 22 × 0.4375 × 4 V = 22 × 1.75 V = 38.5 m³ Step 4: Convert volume to kilolitres. We know that 1 m³ = 1 kilolitre. So, 38.5 m³ = 38.5 kilolitres. Final Answer: The capacity of the conical pit is 38.5 kilolitres.
• Example 2: Find the volume of a right circular cone with radius 6 cm and slant height 10 cm. A: Step 1: Identify given values. Given: Radius r = 6 cm, Slant height l = 10 cm. Goal: Find the volume of the cone. Note that height (h) is not directly given, but slant height (l) is. Step 2: Calculate the height (h) using the Pythagorean theorem. We know l² = r² + h² 10² = 6² + h² 100 = 36 + h² h² = 100 - 36 h² = 64 h = √64 h = 8 cm Step 3: Apply the volume formula. Volume of cone V = (1/3)πr²h V = (1/3) × (22/7) × (6)² × 8 V = (1/3) × (22/7) × 36 × 8 V = (22/7) × 12 × 8 (Cancel 3 with 36) V = (22/7) × 96 V = 2112 / 7 V ≈ 301.71 cm³ Final Answer: The volume of the cone is approximately 301.71 cm³.

Common Mistakes & Exam Tips for Cone Problems

To score well in problems involving the volume of cones, be mindful of these common pitfalls and adopt these smart strategies:

1. Distinguish between Height (h) and Slant Height (l): This is the most frequent mistake. Students often confuse the given 'depth' or 'altitude' (actual height, h) with 'slant height' (l) or vice-versa. Always read the question carefully and draw a small sketch if needed to clarify which dimension is provided.
2. Pythagorean Theorem Application: Remember l² = r² + h². If you are given r and l, you must first find h before you can calculate the volume. Similarly, if h and l are given, you'll need to find r.
3. Units Consistency: Ensure all dimensions (radius, height, slant height) are in the same unit before starting calculations. If diameter is given, convert it to radius (r = d/2). Convert units (e.g., cm to m, or vice versa) at the beginning of the problem, not in the middle or at the end, to avoid errors.
4. Value of Pi (π): Use π = 22/7 unless the question specifically states to use π = 3.14. Using the wrong approximation can lead to slightly different final answers, which might be marked incorrect in some contexts.
5. Calculation Accuracy: Volume formulas involve squares (r²) and multiplication. Double-check your arithmetic, especially when dealing with fractions or decimals. Simplify expressions where possible before multiplying to reduce errors.
6. Final Unit: Always remember to write the correct unit for your final answer. Volume is measured in cubic units (e.g., cm³, m³, litres, kilolitres). Be precise!

Practice Questions with Solutions

• Q: The height of a cone is 15 cm. If its volume is 1570 cm³, find the radius of the base. (Use π = 3.14) A: Step 1: Write down the given values and the formula. Given: Height h = 15 cm, Volume V = 1570 cm³, π = 3.14. Formula: V = (1/3)πr²h. Step 2: Substitute the given values into the formula. 1570 = (1/3) × 3.14 × r² × 15 Step 3: Simplify and solve for r². 1570 = 3.14 × r² × (15/3) 1570 = 3.14 × r² × 5 1570 = 15.7 × r² r² = 1570 / 15.7 r² = 100 Step 4: Find the radius r. r = √100 r = 10 cm Final answer: The radius of the base is 10 cm.
• Q: If the volume of a right circular cone of height 9 cm is 48π cm³, find the diameter of its base. A: Step 1: List the knowns and the formula. Given: Height h = 9 cm, Volume V = 48π cm³. Formula: V = (1/3)πr²h. Step 2: Substitute values into the volume formula and solve for r². 48π = (1/3) × π × r² × 9 48π = π × r² × (9/3) 48π = π × r² × 3 Divide both sides by π: 48 = 3 × r² r² = 48 / 3 r² = 16 Step 3: Find the radius r, then the diameter. r = √16 r = 4 cm Diameter d = 2 × r = 2 × 4 = 8 cm. Final answer: The diameter of the base is 8 cm.
• Q: A heap of wheat is in the form of a cone whose diameter is 10.5 m and height is 3 m. Find its volume. The heap is to be covered by canvas to protect it from rain. Find the area of the canvas required. (Assume π = 22/7) A: Step 1: Identify given dimensions for volume calculation. Given: Diameter = 10.5 m, so Radius r = 10.5 / 2 = 5.25 m. Height h = 3 m. Goal (Part 1): Find volume (capacity) of the heap. Step 2: Calculate the volume of the heap. Volume V = (1/3)πr²h V = (1/3) × (22/7) × (5.25)² × 3 V = (1/3) × (22/7) × (5.25 × 5.25) × 3 V = (22/7) × (5.25 × 5.25) (Cancel 3 from numerator and denominator) V = 22 × (0.75 × 5.25) (Since 5.25/7 = 0.75) V = 22 × 3.9375 V = 86.625 m³ Step 3: Identify dimensions for canvas area (Curved Surface Area). Goal (Part 2): Find the area of canvas required, which is the Curved Surface Area (CSA) of the cone. To find CSA, we need slant height l. l² = r² + h² l² = (5.25)² + 3² l² = 27.5625 + 9 l² = 36.5625 l = √36.5625 ≈ 6.046 m (approximately) Step 4: Calculate the Curved Surface Area (CSA). CSA = πrl CSA = (22/7) × 5.25 × 6.046 CSA = 22 × 0.75 × 6.046 CSA = 16.5 × 6.046 CSA ≈ 99.759 m² Final answer: The volume of the heap is 86.625 m³. The area of the canvas required is approximately 99.76 m² (rounded to two decimal places).

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