NCERT Solutions Class 9 Maths Surface Area and Volume Ex 13.3

Understanding the Geometry of a Right Circular Cone

A right circular cone is a solid generated by revolving a right-angled triangle about one of its sides containing the right angle. It has a flat circular base and a curved surface tapering to a single point called the vertex.

Unlike cylinders or cuboids, cones introduce a unique dimension: the slant height ($l$). The slant height is the distance from the vertex to any point on the circular boundary of the base. We can visualize a right-angled triangle inside the cone where the vertical height ($h$) is the perpendicular altitude, the base radius ($r$) is the base of the triangle, and the slant height ($l$) is the hypotenuse. Using the Pythagoras theorem, we establish the fundamental relationship:

$l^2 = r^2 + h^2$ or $l = \sqrt{r^2 + h^2}$

When we talk about the Curved Surface Area (CSA), we are looking only at the curved canvas part of the cone (excluding the circular bottom). The formula for CSA is:

$\text{CSA} = \pi r l$

If we need to calculate the Total Surface Area (TSA), we must add the flat circular base area ($\pi r^2$) to the curved surface area:

$\text{TSA} = \pi r l + \pi r^2 = \pi r (l + r)$

Mastering these algebraic relationships is the absolute key to scoring full marks in Exercise 13.3 of your NCERT textbook.

Essential Terms and Formulas

Slant Height (l)

The distance measured along the curved side from the apex to the edge of the circular base. Calculated as √(r² + h²).

Curved Surface Area (CSA)

The surface area of only the sloped, curved wall of the cone, excluding the circular base. Formula: πrl.

Total Surface Area (TSA)

The sum of the area of the curved surface and the area of the circular base. Formula: πr(l + r).

Vertical Height (h)

The perpendicular line segment drawn from the vertex of the cone to the center of its circular base.

Step-by-Step Blueprint to Solve Cone Surface Area Problems

1. Step 1: Identify Given Dimensions — Read the problem carefully to identify what parameters are given (radius $r$, diameter $d$, vertical height $h$, or slant height $l$). If diameter is given, divide it by 2 to get the radius.
2. Step 2: Check or Calculate Slant Height ($l$) — If the problem provides vertical height $h$ and radius $r$, do not directly plug $h$ into the formula $\pi r l$. First use Pythagoras' theorem $l = \sqrt{r^2 + h^2}$ to find $l$.
3. Step 3: Choose the Correct Formula — Determine what the question asks. If it asks for canvas needed, painting, or 'curved area', use CSA = $\pi r l$. If it asks for the 'total surface area' or a closed conical container, use TSA = $\pi r (l + r)$.
4. Step 4: Execute Calculations and Append Units — Unless specified otherwise, use $\pi = 22/7$. Always simplify numbers before multiplying out to save time. State your final answer in square units (e.g., $cm^2$ or $m^2$).

Common Board Exam Mistakes & Tips

• The Height Trap: The most frequent mistake students make in Class 9 exams is replacing slant height ($l$) with vertical height ($h$) in the CSA formula. Always double-check if the word used is 'height' ($h$) or 'slant height' ($l$).
• Unit Consistency: Ensure that radius and height/slant height are in the same units (either both in cm or both in m) before executing your calculations. If the cost is given in Rs per $m^2$, convert all dimensions to meters first.
• Simplifying with 7: The value $\pi = 22/7$ is strategically chosen by NCERT paper setters so that either $r$ or $l$ is a multiple of 7. Always look to cancel terms out before doing heavy multiplication.

Practice Questions with Solutions

• Q: Find the curved surface area of a cone if its slant height is 10 cm and the radius of its base is 7 cm. (Take pi = 22/7) A: Step 1: Write down the given values. Radius (r) = 7 cm Slant height (l) = 10 cm Step 2: Apply the formula for Curved Surface Area (CSA). CSA = π r l Step 3: Substitute the values. CSA = (22/7) 7 10 Step 4: Simplify. CSA = 22 * 10 = 220 cm² Final answer: The curved surface area of the cone is 220 cm².
• Q: Find the total surface area of a cone, if its slant height is 21 m and the diameter of its base is 24 m. A: Step 1: Write down the given values and find the radius. Slant height (l) = 21 m Diameter = 24 m Radius (r) = diameter / 2 = 24 / 2 = 12 m Step 2: Apply the formula for Total Surface Area (TSA). TSA = π r (l + r) Step 3: Substitute the values. TSA = (22/7) 12 (21 + 12) TSA = (22/7) 12 33 Step 4: Perform the multiplication. TSA = (22 12 33) / 7 TSA = 8712 / 7 ≈ 1244.57 m² Final answer: The total surface area of the cone is 1244.57 m² (or 1244 4/7 m²).
• Q: Curved surface area of a cone is 308 cm² and its slant height is 14 cm. Find (i) radius of the base and (ii) total surface area of the cone. A: Step 1: Solve for radius (r) using the CSA formula. Given, CSA = 308 cm², Slant height (l) = 14 cm CSA = π r l 308 = (22/7) r 14 308 = 22 r 2 308 = 44 r r = 308 / 44 r = 7 cm Step 2: Calculate the Total Surface Area (TSA) using the computed radius. TSA = π r (l + r) TSA = (22/7) 7 (14 + 7) TSA = 22 21 TSA = 462 cm² Final answer: (i) Radius of the base is 7 cm. (ii) Total surface area of the cone is 462 cm².
• Q: A conical tent is 10 m high and the radius of its base is 24 m. Find (i) slant height of the tent, (ii) cost of the canvas required to make the tent, if the cost of 1 m² canvas is Rs 70. A: Step 1: Find the slant height (l). Given, height (h) = 10 m, radius (r) = 24 m Using Pythagoras theorem: l² = r² + h² l² = 24² + 10² l² = 576 + 100 l² = 676 l = √676 = 26 m Step 2: Find the Curved Surface Area (CSA) of the tent (since a tent only covers the sides, excluding the ground base). CSA = π r l CSA = (22/7) 24 26 CSA = 13728 / 7 m² Step 3: Calculate the total cost. Cost of 1 m² canvas = Rs 70 Total Cost = CSA Rate Total Cost = (13728 / 7) 70 Total Cost = 13728 * 10 = Rs 1,37,280 Final answer: (i) Slant height of the tent is 26 m. (ii) Cost of the canvas required to make the tent is Rs 1,37,280.

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