Curved Surface Area of Cone with Slant Height Examples

Curved Surface Area of Cone is the area of the sloping outer surface of a cone, excluding its circular base. It is calculated using the radius and slant height. This page covers the curved surface area of cone formula, methods for finding missing values, unit conversions, solved examples and common aptitude problems.

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What Is the Curved Surface Area of a Cone?

The curved surface area of a cone is the area of its curved lateral surface, excluding the circular base. Its formula is πrl, where r is the base radius and l is the slant height.

For a cone with radius r and slant height l, curved surface area = πrl square units. The slant height is measured along the curved side from the vertex to the edge of the circular base. If vertical height h is given, first calculate l using l = √(r² + h²).

Curved Surface Area of Cone Video Lessons

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Cone Volume and Surface Area

Learn the formulas for a cone’s volume, slant height, curved surface area, and total surface area, with guidance on applying them to mensuration problems.

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Curved Surface Area of Cone Formula & Tricks

Important Formulas

Curved surface area
CSA = πrl

Here, r is the radius of the circular base and l is the slant height.

Slant height
l = √(r² + h²)

For a right circular cone, h is the perpendicular height and r is the base radius.

Diameter form
CSA = πdl/2

Since r = d/2, the formula can be written using diameter d.

Using circumference
CSA = (1/2) × circumference of base × l

The curved surface unfolds into a sector whose arc length equals the circumference of the base.

Quick Tricks

Use the diameter directly

When diameter is given, use CSA = πdl/2 instead of first dividing the diameter by 2.

Example: If d = 14 cm and l = 10 cm, CSA = (22/7) × 14 × 10 ÷ 2 = 220 cm².
Find slant height before applying the formula

If only radius and perpendicular height are given, use the right-triangle relation l = √(r² + h²).

Example: For r = 6 cm and h = 8 cm, l = √(36 + 64) = 10 cm. Therefore, CSA = π × 6 × 10 = 60π cm².
Keep units consistent

Convert radius, slant height and height to the same unit before calculating area. The final unit is squared.

Example: If r = 0.5 m and l = 80 cm, convert l to 0.8 m. Then CSA = π × 0.5 × 0.8 = 0.4π m².

Curved Surface Area of Cone Concepts

Radius and Slant Height in a Cone

The radius is the distance from the centre of the circular base to its boundary, while the slant height is the distance from the vertex to a point on the boundary of the base.

In a right circular cone, the radius r, perpendicular height h and slant height l form a right triangle. Therefore, l² = r² + h². The perpendicular height is not used directly in CSA = πrl unless it is first used to find l.

Example: If r = 5 cm and h = 12 cm, then l = √(5² + 12²) = √169 = 13 cm.

Derivation from the Unfolded Cone

When the curved surface of a cone is opened out, it forms a sector of a circle with radius l and sector arc length 2πr.

The area of a sector is one-half × radius × arc length. Hence, curved surface area = 1/2 × l × 2πr = πrl. The circular base is not included in this calculation.

Example: For r = 7 cm and l = 9 cm, CSA = π × 7 × 9 = 63π cm².

Calculating CSA When Height Is Given

If the perpendicular height is given instead of the slant height, calculate the slant height using the Pythagorean theorem before finding CSA.

Use l = √(r² + h²), then substitute l in πrl. For a right circular cone, this method applies because the radius and perpendicular height are perpendicular sides of a right triangle.

Example: For r = 9 cm and h = 12 cm, l = √(81 + 144) = 15 cm. Thus, CSA = π × 9 × 15 = 135π cm².

Difference Between CSA and Total Surface Area

Curved surface area excludes the circular base, whereas total surface area includes the curved surface and the base.

CSA = πrl. Total surface area = πrl + πr² = πr(l + r). Use the total surface area formula only when the question includes the base of the cone.

Example: For r = 7 cm and l = 10 cm, CSA = 70π cm², while total surface area = 70π + 49π = 119π cm².

Curved Surface Area of Cone Visual Guide

Use the diagram or chart below to understand the concept clearly.

Cone showing radius and slant height

Visual guide: Cone showing radius and slant height
Radius: r = 3 cm Slant height: l = 5 cm Curved surface area: CSA = 15π cm²
The curved surface area uses the base radius r and slant height l; the circular base is excluded.

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Practice Curved Surface Area of Cone Questions

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1The price of petrol is increased by 28%. A person wants to increase his expenditure by 22% only. By approximately what percent should he decrease his consumption?→ 2An article is sold for Rs. 288 after successive discounts of 20% and 25%. What is the marked price of the article?→ 321.6 ÷ 3.6 × 2 + 0.25 × 16 ÷ 4 - 6 is equal to∶→ 4What is the value of x so that the seven digit number 55350x2 is divisible by 72?→ 5Two articles are sold for Rs. 4,752 each. On one, the seller gains 32% and on the other he loses 28%. What is his overall gain or loss percentage, correct to one decimal place?→ 6The price of sugar is increased by 17%. A person wants to increase his expenditure by 5% only. By approximately what percent should he decrease his consumption?→ 7The value of 15.2 + 5.8 ÷ 2.9 × 2 - 3.5 × 2 ÷ 0.5 is equal to:→ 8What is the value of x so that the seven digit number 8439x53 is divisible by 99?→ 9Two articles are sold for Rs. 962 each. On one, the seller gains 30% and on the other he loses 26%. What is his overall gain or loss percentage, nearest to one decimal place?→ 10An article is sold for Rs. 288 after successive discounts of 25% and x%. If the marked price of the article is Rs. 480, what is the value of x?→

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Quick Revision Notes

Curved Surface Area of Cone: Revision Points

Use these formulas and rules to revise cone curved-surface calculations.

  • CSA of a cone = πrl.
  • r is the base radius and l is the slant height.
  • If diameter d is given, CSA = πdl/2.
  • If perpendicular height h is given, first calculate l = √(r² + h²).
  • CSA excludes the circular base; total surface area includes it and equals πr(l + r).
  • Use the same unit for r and l, and express the answer in square units.

Related Topic Explainers

Area of Square → Perimeter of Square → Area of Rectangle → Perimeter of Rectangle → Area of Triangle → Perimeter of Triangle → Heron's Formula → Area of Triangle Using Two Sides and Included Angle →

Curved Surface Area of Cone FAQs

What is the curved surface area of a cone formula?

The formula is CSA = πrl, where r is the radius of the base and l is the slant height.

How do you find the curved surface area when diameter is given?

Use CSA = πdl/2, where d is the diameter and l is the slant height. For d = 14 cm and l = 10 cm, CSA = 220 cm² using π = 22/7.

How is slant height calculated from radius and height?

For a right circular cone, l = √(r² + h²). If r = 6 cm and h = 8 cm, then l = √100 = 10 cm.

What is the curved surface area of a cone with radius 7 cm and slant height 12 cm?

CSA = π × 7 × 12 = 84π cm². Using π = 22/7, the area is 264 cm².

What is the difference between curved surface area and total surface area of a cone?

Curved surface area is πrl and excludes the base. Total surface area is πrl + πr², or πr(l + r), and includes the circular base.

What happens to the curved surface area if the radius is doubled while slant height remains constant?

The curved surface area also doubles because CSA = πrl is directly proportional to r when l is constant.

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