Slant Height of Cone with Pythagoras Formula

Slant Height of Cone is the distance from the vertex to any point on the circular edge of the base, measured along the curved surface. For a right circular cone, it is found using the radius and perpendicular height. This page covers the formula, solved calculations, surface-area applications, and useful aptitude shortcuts.

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What Is the Slant Height of a Cone?

The slant height of a right circular cone is the straight-line distance from its vertex to a point on the circumference of its circular base. It is usually denoted by l.

The radius r, perpendicular height h, and slant height l form a right-angled triangle. Therefore, l² = r² + h², so l = √(r² + h²). Slant height is measured in the same unit as r and h. For example, if r = 5 cm and h = 12 cm, then l = √(25 + 144) = 13 cm.

Slant Height 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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Slant Height of Cone Formula & Tricks

Important Formulas

Slant height formula
l = √(r² + h²)

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

Perpendicular height
h = √(l² − r²)

Use this form when the slant height and radius are given.

Radius from slant height
r = √(l² − h²)

Use this form when the slant height and perpendicular height are known.

Curved surface area using slant height
CSA = πrl

The curved surface area of a cone is π times its radius times its slant height.

Total surface area using slant height
TSA = πr(l + r)

Add the base area πr² to the curved surface area πrl.

Quick Tricks

Recognise Pythagorean triples

If the radius and height form a known right-triangle triple, the slant height can be obtained directly. Common triples include 3-4-5, 5-12-13, and 7-24-25.

Example: For r = 7 cm and h = 24 cm, l = 25 cm because 7² + 24² = 25².
Check that the slant height is the longest length

In a right circular cone, l² = r² + h². Hence, the slant height must be greater than both the radius and the perpendicular height.

Example: If r = 6 cm and h = 8 cm, l = 10 cm, which is greater than 6 cm and 8 cm.
Keep units consistent before squaring

Convert radius and height to the same unit before applying the formula. The final slant height has that same unit.

Example: If r = 0.3 m and h = 40 cm, first convert 0.3 m to 30 cm. Then l = √(30² + 40²) = 50 cm.

Slant Height of Cone Concepts

Slant Height from Radius and Perpendicular Height

Calculate the slant height by applying the Pythagorean theorem to the triangle formed by r, h, and l.

In a vertical cross-section through the axis of a right cone, the radius is perpendicular to the height. Thus, l² = r² + h². Substitute the given values, add the squares, and take the positive square root because length cannot be negative.

Example: For r = 9 cm and h = 12 cm, l = √(9² + 12²) = √225 = 15 cm.

Finding a Missing Radius or Height

When l and one other dimension are given, subtract the square of the known shorter dimension from l² and take the square root.

Use r = √(l² − h²) to find the radius or h = √(l² − r²) to find the perpendicular height. The slant height must be greater than the dimension being subtracted.

Example: If l = 13 cm and r = 5 cm, h = √(13² − 5²) = √144 = 12 cm.

Slant Height in Curved Surface Area

The curved surface area of a cone uses the slant height, not the perpendicular height: CSA = πrl.

The curved surface is formed by the sector of a circle whose radius is l. If r = 7 cm and l = 25 cm, then CSA = π × 7 × 25 = 175π cm², or 550 cm² when π = 22/7.

Example: For r = 7 cm and l = 25 cm, CSA = 175π cm² = 550 cm² using π = 22/7.

Difference Between Slant Height and Perpendicular Height

Perpendicular height is the vertical distance from the vertex to the centre of the base, whereas slant height reaches the circular edge along the surface.

They are different lengths unless the radius is zero. For a right cone, l² = h² + r², so l is always greater than h when r > 0.

Example: For r = 5 cm and h = 12 cm, the perpendicular height is 12 cm but the slant height is 13 cm.

Slant Height of Cone Visual Guide

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

Right circular cone showing slant height

Visual guide: Right circular cone showing slant height
Radius r: 7 cm Perpendicular height h: 24 cm Slant height l: 25 cm
The radius, perpendicular height, and slant height form a right triangle with l² = r² + h².

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Practice Slant Height of Cone 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

Slant Height of Cone: Revision Points

Use these formulas and checks for quick revision.

  • For a right circular cone, l = √(r² + h²).
  • The slant height is measured from the vertex to the circumference of the base.
  • The perpendicular height goes from the vertex to the centre of the base at 90°.
  • To find height, use h = √(l² − r²).
  • To find radius, use r = √(l² − h²).
  • Curved surface area is πrl, while total surface area is πr(l + r).
  • Use the same unit for radius and height before applying the formula.
  • The slant height is greater than the perpendicular height and radius for a non-degenerate cone.

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 →

Slant Height of Cone FAQs

What is the slant height of a cone formula?

For a right circular cone, the formula is l = √(r² + h²), where r is the radius and h is the perpendicular height.

How do you find the slant height when the radius is 8 cm and height is 15 cm?

l = √(8² + 15²) = √289 = 17 cm.

Can the perpendicular height be used directly in the curved surface area formula?

No. The curved surface area is CSA = πrl, so it requires the slant height l, not the perpendicular height h.

How do you find the height if the slant height is 10 cm and the radius is 6 cm?

h = √(10² − 6²) = √64 = 8 cm.

What is the slant height when the radius is 3 cm and the height is 4 cm?

l = √(3² + 4²) = √25 = 5 cm.

Is the slant height always greater than the perpendicular height?

For a non-degenerate right cone with positive radius, yes. Since l² = h² + r², l > h.

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