Surface Area of Sphere Calculated Using Radius

Surface Area of Sphere means the total area covering the outside of a spherical solid. Its formula is 4πr², where r is the radius. This page explains the formula, diameter-based calculation, unit conversion, radius changes, shortcuts and solved surface area of sphere problems for quantitative aptitude.

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What Is the Surface Area of a Sphere?

The surface area of a sphere is the area of its complete curved outer surface. It is calculated using 4πr², where r is the radius of the sphere.

A sphere has no separate flat base or curved-surface parts, so its total surface area is also its curved surface area. The answer is expressed in square units, such as cm², m² or mm². For a sphere with radius 7 cm, surface area = 4 × 22/7 × 7² = 616 cm².

Surface Area of Sphere Video Lessons

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Sphere and Hemisphere: Volume and Surface Area

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Surface Area of Sphere Formula & Tricks

Important Formulas

Surface area using radius
S = 4πr²

Here, S is the surface area and r is the radius of the sphere.

Surface area using diameter
S = πd²

Since d = 2r, substituting r = d/2 in 4πr² gives S = πd².

Radius from surface area
r = √(S ÷ 4π)

Use this form when the surface area is given and the radius is required.

Change in surface area
S₂ ÷ S₁ = (r₂ ÷ r₁)²

For spheres made of the same type of measurement, surface area is proportional to the square of the radius.

Quick Tricks

Use the diameter shortcut

When the diameter is given directly, use S = πd² instead of first finding the radius.

Example: If d = 14 cm, S = 22/7 × 14² = 616 cm².
Square the radius before multiplying

Calculate r² first and then multiply by 4π. This reduces calculation errors in numerical questions.

Example: For r = 3.5 cm, r² = 12.25, so S = 4 × 22/7 × 12.25 = 154 cm².
Track radius changes as squares

If the radius becomes k times, the surface area becomes k² times. A doubled radius gives four times the surface area.

Example: If the radius changes from 5 cm to 10 cm, the surface area ratio is (10/5)² = 4.

Surface Area of Sphere Concepts

Radius and Diameter in the Formula

The radius is half the diameter, so the surface area can be calculated as 4πr² or πd².

Use r when the radius is given. If the diameter d is given, either divide it by 2 and use 4πr², or directly use πd². The diameter must be measured in the same unit as the radius.

Example: For a diameter of 10 cm, S = π × 10² = 100π cm² = 314.29 cm² approximately.

Calculating Surface Area with π

Use π = 22/7 when the radius or diameter is a suitable multiple of 7; otherwise, π = 3.14 or the exact form π may be used as instructed.

The surface area remains in square units. If the radius is in metres, the result is in m². For r = 7 m and π = 22/7, S = 4 × 22/7 × 49 = 616 m².

Example: For r = 3 cm using π = 3.14, S = 4 × 3.14 × 9 = 113.04 cm².

Effect of Changing the Radius

The surface area changes in the square ratio of the radii.

If the radius changes from r₁ to r₂, then S₂/S₁ = r₂²/r₁². Therefore, a 10% increase in radius does not produce a 10% increase in surface area; the new area is (1.1)² = 1.21 times the original area, or 21% greater.

Example: If the radius increases from 10 cm to 15 cm, the surface area becomes (15/10)² = 2.25 times the original.

Finding Radius from Surface Area

To find the radius when surface area is known, use r = √(S/(4π)).

First divide the given surface area by 4π, then take the positive square root because a radius cannot be negative. If S = 616 cm² and π = 22/7, r = √(616 ÷ (4 × 22/7)) = √49 = 7 cm.

Example: A sphere with surface area 154 cm² has radius √(154 ÷ (4 × 22/7)) = √(12.25) = 3.5 cm.

Surface Area of Sphere Visual Guide

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

Sphere with radius marked

Visual guide: Sphere with radius marked
Radius: r Surface area: 4πr²
The surface area of the complete spherical surface is 4πr².

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Practice Surface Area of Sphere 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

Surface Area of Sphere Revision Points

Remember the formulas, units and radius-based properties used in sphere surface area questions.

  • Surface area of a sphere = 4πr².
  • Using diameter d, surface area = πd².
  • The radius is half the diameter: r = d/2.
  • Surface area is measured in square units.
  • If radius becomes k times, surface area becomes k² times.
  • For a given surface area, radius = √(S/(4π)).
  • A sphere has only one complete curved surface, so its total surface area equals its 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 →

Surface Area of Sphere FAQs

What is the surface area of sphere formula?

The formula is S = 4πr², where r is the radius. In terms of diameter d, it is S = πd².

What is the surface area of a sphere with radius 7 cm?

S = 4 × 22/7 × 7² = 616 cm².

What is the surface area of a sphere with diameter 14 cm?

Using S = πd², S = 22/7 × 14² = 616 cm².

What happens to the surface area if the radius is doubled?

The surface area becomes four times the original because surface area is proportional to r².

A sphere has surface area 616 cm². What is its radius?

Using r = √(S/(4π)), r = √(616 ÷ (4 × 22/7)) = √49 = 7 cm.

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

There is no difference for a complete sphere. Both refer to the area of its entire outer surface and equal 4πr².

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