Total Surface Area of Cone Using Radius and Slant Height

Total Surface Area of Cone is the sum of its curved surface area and circular base area. It is calculated using the radius and slant height. This page explains the total surface area of cone formula, the relation between height and slant height, solved numerical examples, and common aptitude problems involving cones.

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What Is the Total Surface Area of a Cone?

The total surface area of a cone is the area of its curved surface plus the area of its circular base. Its formula is TSA = πr(l + r), where r is the radius and l is the slant height.

For a cone, curved surface area = πrl and base area = πr². Therefore, total surface area = πrl + πr² = πr(l + r). If the perpendicular height h and radius r are given, first find the slant height using l = √(r² + h²). The answer is expressed in square units.

Total 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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Total Surface Area of Cone Formula & Tricks

Important Formulas

Total surface area
TSA = πr(l + r)

Use this formula when the radius r and slant height l are known.

Curved surface area
CSA = πrl

This is the area of the curved part of the cone, excluding its circular base.

Base area
Base area = πr²

The base of a cone is a circle with radius r.

Slant height
l = √(r² + h²)

Use this relation when the radius r and perpendicular height h are given.

Quick Tricks

Add the base area to the curved area

Do not use πrl alone for total surface area. Add the circular base area πr², giving TSA = πr(l + r).

Example: If r = 7 cm and l = 10 cm, TSA = π × 7 × (10 + 7) = 374 cm² using π = 22/7.
Find slant height before applying TSA

When height and radius are given, use the right triangle inside the cone: l² = h² + r². Then substitute l in πr(l + r).

Example: For r = 5 cm and h = 12 cm, l = √(5² + 12²) = √169 = 13 cm.
Use the suitable value of π

Use π = 22/7 when the calculation cancels 7 conveniently. Otherwise, use π = 3.14 or leave the answer in terms of π if no approximation is requested.

Example: For r = 7 cm and l = 10 cm, πr(l + r) becomes (22/7) × 7 × 17 = 374 cm².

Total Surface Area of Cone Concepts

Curved Surface Area and Base Area

The total surface area is obtained by adding the curved surface area and one circular base area.

The curved surface area is πrl, while the base area is πr². Hence, TSA = πrl + πr² = πr(l + r). The base is counted once because a cone has one circular base.

Example: For r = 3 cm and l = 5 cm, CSA = 15π cm², base area = 9π cm², and TSA = 24π cm².

Using Height to Find Slant Height

If perpendicular height and radius are given instead of slant height, calculate l using l = √(h² + r²).

The radius, height and slant height form a right triangle. The slant height is the hypotenuse, so l² = h² + r². After finding l, apply TSA = πr(l + r).

Example: For r = 5 cm and h = 12 cm, l = √(25 + 144) = 13 cm. Therefore, TSA = π × 5 × (13 + 5) = 90π cm².

Calculation When Diameter Is Given

When diameter d is given, first convert it to radius using r = d/2.

Substitute r = d/2 in the total surface area formula. Thus, TSA = π(d/2)(l + d/2), provided the slant height is known.

Example: If d = 14 cm and l = 10 cm, r = 7 cm. Therefore, TSA = π × 7 × 17 = 119π cm² = 374 cm² using π = 22/7.

Units and Comparison of Areas

Total surface area is measured in square units such as cm², m² or mm².

All lengths must be converted to the same unit before calculation. If every linear dimension is multiplied by k, the total surface area is multiplied by k².

Example: A cone with TSA 40 cm² has TSA 40 × 100² = 400,000 cm² when all dimensions are scaled 100 times.

Total Surface Area of Cone Visual Guide

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

Cone Showing Radius, Height and Slant Height

Visual guide: Cone Showing Radius, Height and Slant Height
Radius: r Perpendicular height: h Slant height: l = √(r² + h²)
The radius r, perpendicular height h and slant height l form a right triangle inside the cone.

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Practice Total Surface Area 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

Total Surface Area of Cone: Revision Points

Recall these formulas and calculation rules for cone surface-area problems.

  • TSA = curved surface area + base area.
  • Curved surface area = πrl.
  • Base area = πr².
  • Total surface area = πr(l + r).
  • If height and radius are given, l = √(h² + r²).
  • If diameter d is given, radius r = d/2.
  • Use square units for surface area.
  • Convert all lengths to the same unit before substituting values.

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 →

Total Surface Area of Cone FAQs

What is the total surface area of a cone formula?

The formula is TSA = πr(l + r), where r is the radius and l is the slant height.

What is the total surface area of a cone with radius 7 cm and slant height 10 cm?

TSA = π × 7 × (10 + 7) = 119π cm². Using π = 22/7, the area is 374 cm².

How is the slant height of a cone calculated?

If radius r and perpendicular height h are known, slant height l = √(r² + h²). For r = 5 cm and h = 12 cm, l = 13 cm.

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 includes the circular base, so it is πrl + πr² = πr(l + r).

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

First find the radius using r = d/2, then use TSA = πr(l + r). For diameter 14 cm and slant height 10 cm, TSA = 374 cm² using π = 22/7.

What is the unit of total surface area of a cone?

It is expressed in square units, such as cm², m² or mm², because surface area is a two-dimensional measurement.

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