Curved Surface Area of Frustum Using Slant Height

Curved Surface Area of Frustum is the area of its sloping lateral surface, excluding the two circular bases. It is calculated using the radii of the two circular ends and the frustum’s slant height. This page covers the standard formula, slant-height calculation, diameter form, solved examples, and common mensuration questions.

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What Is the Curved Surface Area of a Frustum?

The curved surface area of a frustum is the area of its curved lateral surface between two parallel circular bases. For a frustum with larger radius R, smaller radius r, and slant height l, CSA = π(R + r)l.

The slant height is measured along the inclined surface, not vertically. If the perpendicular height is h, then l = √[h² + (R − r)²]. Therefore, the complete formula is CSA = π(R + r)√[h² + (R − r)²]. The answer is expressed in square units.

Curved Surface Area of Frustum Video Lessons

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Frustum Volume and Lateral Surface Area

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Curved Surface Area of Frustum Formula & Tricks

Important Formulas

Curved surface area
CSA = π(R + r)l

R is the larger base radius, r is the smaller base radius, and l is the slant height.

Slant height
l = √[h² + (R − r)²]

h is the perpendicular height of the frustum. The difference of the radii forms the horizontal side of the right triangle.

CSA using perpendicular height
CSA = π(R + r)√[h² + (R − r)²]

Use this form when the perpendicular height and both radii are given.

CSA using diameters
CSA = (π/2)(D + d)l

D and d are the larger and smaller base diameters, respectively, because R = D/2 and r = d/2.

Quick Tricks

Use the diameter shortcut directly

When diameters are given, do not first convert both to radii. Substitute them in CSA = (π/2)(D + d)l.

Example: If D = 14 cm, d = 6 cm and l = 5 cm, CSA = (π/2)(20)(5) = 50π cm².
Check the slant height before calculating area

The slant height must be greater than or equal to the perpendicular height. Calculate l from the right triangle whenever it is not given.

Example: For h = 4 cm, R = 7 cm and r = 3 cm, l = √(4² + 4²) = 4√2 cm, not 5 cm. A value of 5 cm applies only when the radius difference is 3 cm.

Curved Surface Area of Frustum Concepts

Calculating Slant Height from Height and Radii

The slant height of a frustum is the hypotenuse of a right triangle with perpendicular side h and horizontal side R − r.

Use l = √[h² + (R − r)²]. The difference between the radii is taken as a positive length, so (R − r)² gives the same result even if the radii are listed in reverse order.

Example: If h = 12 cm, R = 10 cm and r = 5 cm, then l = √(12² + 5²) = √169 = 13 cm.

Applying the Curved Surface Area Formula

After finding l, multiply π by the sum of the two radii and the slant height: CSA = π(R + r)l.

All lengths must use the same unit before substitution. If π = 22/7 is used, ensure that the radii and slant height are compatible with the required answer unit.

Example: For R = 10 cm, r = 5 cm and l = 13 cm, CSA = π(10 + 5)(13) = 195π cm².

Distinguishing Curved and Total Surface Area

Curved surface area excludes both circular bases, while total surface area includes the two base areas.

For a frustum, TSA = π(R + r)l + πR² + πr². Thus, TSA = π[(R + r)l + R² + r²]. Do not add the base areas when the question asks only for curved surface area.

Example: If CSA = 195π cm², R = 10 cm and r = 5 cm, TSA = 195π + 100π + 25π = 320π cm².

Using Diameters Instead of Radii

When the base diameters are given, use CSA = (π/2)(D + d)l.

This formula follows from R = D/2 and r = d/2. It avoids converting each diameter separately and is valid only when D and d refer to the larger and smaller circular bases.

Example: For D = 14 cm, d = 6 cm and l = 5 cm, CSA = (π/2)(14 + 6)(5) = 50π cm².

Curved Surface Area of Frustum Visual Guide

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

Frustum dimensions used in the CSA formula

Visual guide: Frustum dimensions used in the CSA formula
Larger radius: R Smaller radius: r Perpendicular height: h Slant height: l
A frustum has larger radius R, smaller radius r, perpendicular height h, and slant height l.

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Practice Curved Surface Area of Frustum Questions

Practise published questions related to this topic.

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 Frustum: Revision Points

Use these formulas and checks for quick revision.

  • CSA of a frustum = π(R + r)l.
  • Slant height l = √[h² + (R − r)²].
  • The curved surface area does not include either circular base.
  • If diameters are given, CSA = (π/2)(D + d)l.
  • Use square units for the final area.
  • For total surface area, add πR² + πr² to the curved surface area.

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 Frustum FAQs

What is the formula for the curved surface area of a frustum?

The formula is CSA = π(R + r)l, where R and r are the radii of the larger and smaller bases and l is the slant height.

How is the slant height of a frustum calculated?

Use l = √[h² + (R − r)²], where h is the perpendicular height and R − r is the difference between the base radii.

What is the curved surface area when R = 7 cm, r = 3 cm and l = 5 cm?

CSA = π(7 + 3)(5) = 50π cm², approximately 157.08 cm².

Can the curved surface area of a frustum be found from diameters?

Yes. If D and d are the base diameters, use CSA = (π/2)(D + d)l.

Does curved surface area include the circular bases?

No. It includes only the sloping lateral surface. The circular bases are included only in the total surface area.

What is the total surface area formula for a frustum?

TSA = π(R + r)l + πR² + πr². The first term is the curved surface area and the remaining terms are the two base areas.

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