RD Sharma Class 9 Solutions Chapter 20 - Surface Area and Volume of Right Circular Cone (Ex 20.1) Exercise 20.1 - Free PDF
FAQs on RD Sharma Class 9 Solutions Chapter 20 - Exercise 20.1
1. What are the key formulas used in RD Sharma Class 9 Solutions for Chapter 20, Exercise 20.1?
The solutions for Exercise 20.1 primarily rely on formulas related to the surface area of a right circular cone. The main formulas are:
- Slant Height (l): l = √(r² + h²), derived from the Pythagorean theorem, where 'r' is the radius and 'h' is the perpendicular height.
- Curved Surface Area (CSA): CSA = πrl. This formula calculates the area of the conical surface only.
- Total Surface Area (TSA): TSA = πr(l + r), which is the sum of the Curved Surface Area (πrl) and the area of the circular base (πr²).
2. How do you find the slant height of a cone when only the radius and perpendicular height are provided in a problem?
To find the slant height (l) of a right circular cone when given the radius (r) and perpendicular height (h), you must use the Pythagorean theorem. The height, radius, and slant height form a right-angled triangle with the slant height as the hypotenuse. The formula is: l² = r² + h². Therefore, you calculate the slant height by taking the square root: l = √(r² + h²). This value is essential for finding the surface area.
3. What is the correct method to calculate the Curved Surface Area (CSA) of a cone as per the solutions in this chapter?
The step-by-step method to calculate the Curved Surface Area (CSA) is as follows:
1. Identify the given values: radius (r) and slant height (l).
2. If the slant height (l) is not given, calculate it using the formula l = √(r² + h²).
3. Apply the formula for CSA: CSA = πrl.
4. Substitute the values of π (usually 22/7 or 3.14), radius (r), and slant height (l) to get the final answer in square units.
4. How are the calculations for Total Surface Area (TSA) and Curved Surface Area (CSA) different in the problems of Exercise 20.1?
The key difference lies in what part of the cone is being measured.
- Curved Surface Area (CSA) measures only the area of the slanted, curved surface of the cone. Its formula is πrl.
- Total Surface Area (TSA) measures the area of all surfaces, which includes the curved surface and the flat circular base. Its formula is TSA = CSA + Area of Base = πrl + πr².
Therefore, to find TSA, you must always include the area of the circular base in your calculation.
5. If a question in Exercise 20.1 gives the circumference of the cone's base, what is the first step to find its surface area?
If the circumference (C) of the base is given, the first step is to calculate the radius (r), as it is needed for all surface area formulas. The formula for the circumference of a circle is C = 2πr. You must rearrange this to solve for the radius: r = C / (2π). Once you have the radius, you can proceed to find the slant height and then calculate the CSA or TSA as required by the problem.
6. Why is the slant height (l) used to calculate the surface area of a cone instead of its perpendicular height (h)?
The surface area represents the area of the material on the exterior of the cone. The slant height (l) is the actual length from the tip of the cone to any point on the edge of its circular base, measured along the surface. In contrast, the perpendicular height (h) is the internal distance from the tip to the center of the base. Since we are measuring the outer surface, we must use the dimension that lies on that surface, which is the slant height.
7. How does the Pythagorean theorem become the fundamental tool for solving problems in RD Sharma Chapter 20, Exercise 20.1?
The Pythagorean theorem is fundamental because a cone's radius (r), perpendicular height (h), and slant height (l) form a right-angled triangle, with the slant height as the hypotenuse. Many problems in Exercise 20.1 do not provide all three dimensions. The theorem, l² = r² + h², is the essential tool that allows you to calculate a missing dimension (most often the slant height) which is mandatory for applying the surface area formulas like CSA = πrl.
8. What is the conceptual difference between the 'surface area' covered in Exercise 20.1 and the 'volume' of a cone?
The conceptual difference is about dimensionality and what is being measured:
- Surface Area (in Ex 20.1) is a two-dimensional measurement. It represents the total area of the exterior surfaces, like the amount of paper needed to cover the cone. It is measured in square units (e.g., cm²).
- Volume is a three-dimensional measurement. It represents the capacity or the amount of space inside the cone, like the amount of water it can hold. It is measured in cubic units (e.g., cm³).
9. In what real-world scenarios would you need to calculate only the Curved Surface Area (CSA) and not the Total Surface Area (TSA) of a cone?
You would calculate only the Curved Surface Area (CSA) in situations where the circular base is open, not used, or not made of the same material. Common examples include:
- Calculating the amount of canvas needed to make a conical tent, as the floor is open or made of different material.
- Finding the surface area of an ice cream cone (the wafer part), which is hollow.
- Determining the area to be painted on a conical traffic cone or a funnel, where the base is open.
