Surface Area of Combined Solids with Worked Examples

Surface Area of Combined Solids is found by adding only the exposed surfaces of two or more joined solids. The surfaces touching inside the combined figure are not included. This page covers the required formulas, contact-surface rules, calculation methods and solved examples for common mensuration problems.

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What Is the Surface Area of Combined Solids?

The surface area of combined solids is the total area of all outer surfaces of a solid formed by joining two or more separate solids.

First identify every exposed face, curved surface and base. Do not count a surface that lies at the junction of two solids. If complete surface areas are added, subtract twice the common contact area because it was counted once for each solid. For one common contact area C, combined surface area = SA₁ + SA₂ − 2C.

Surface Area of Combined Solids Video Lessons

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Combined Solids: Volume and Surface Area

Learn how to calculate the volume and surface area of combined three-dimensional solids by identifying component shapes and applying the appropriate mensuration formulas.

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Surface Area of Combined Solids Formula & Tricks

Important Formulas

General combined-solid rule
External surface area = Sum of individual total surface areas − 2 × total common contact area

Use this rule when the total surface areas of the separate solids are calculated first. Each internal contact surface must be removed from both solids.

Cuboid
TSA = 2(lb + bh + hl)

Here, l is length, b is breadth and h is height.

Cylinder
CSA = 2πrh; TSA = 2πr(h + r)

r is the radius and h is the height. The total surface area includes both circular bases.

Cone
CSA = πrl; TSA = πr(l + r)

r is the radius and l is the slant height. The total surface area includes the circular base.

Sphere
Surface area = 4πr²

A sphere has only one continuous curved outer surface.

Hemisphere
CSA = 2πr²; TSA = 3πr²

The curved surface area excludes the circular base. The total surface area includes the base.

Quick Tricks

Mark hidden surfaces before calculating

A surface at the junction of two solids is internal and must not be counted. If separate total surface areas are added, subtract twice the junction area.

Example: When a cone is joined to a cylinder through a circular base of radius r, the hidden area is πr². Thus, subtract 2πr² from the sum of their total surface areas.
Use exposed-part addition directly

Instead of calculating complete surface areas and subtracting contacts, add only the visible curved surfaces and outer bases shown in the figure.

Example: For a cylinder with a hemisphere attached to its top, use cylinder curved area + cylinder bottom area + hemisphere curved area.
Check dimensions before using formulas

All lengths must be in the same unit. The final surface area has square units, such as cm² or m².

Example: Convert 1 m to 100 cm before combining it with a radius given as 20 cm.

Surface Area of Combined Solids Concepts

Exposed and Hidden Surfaces

Only surfaces visible on the outside of the combined solid contribute to its external surface area.

A circular base joining a cylinder and a hemisphere becomes an internal surface, so it is excluded. A base that remains uncovered is included. The curved surface of a cylinder, cone or hemisphere is included whenever it forms part of the outside boundary.

Example: For a cylinder joined to a hemisphere at one circular base, the cylinder's top base and the hemisphere's flat base are hidden. The cylinder's bottom base, cylinder's curved surface and hemisphere's curved surface remain exposed.

Cylinder Joined to a Hemisphere

For a cylinder of radius r and height h joined to a hemisphere of the same radius, external surface area = 2πrh + πr² + 2πr² = 2πrh + 3πr².

The exposed parts are the cylinder's curved surface, its free circular base and the hemisphere's curved surface. The circular junction is not counted. Therefore, the result is also 2πr(h + 3r/2).

Example: If r = 3 cm and h = 8 cm, external area = 2π(3)(8) + π(3²) + 2π(3²) = 48π + 9π + 18π = 75π cm².

Cylinder Joined to a Cone

For a cone joined to a cylinder through a circular base of radius r, external surface area = πrl + 2πrh + πr², where l is the cone's slant height.

The cone's curved surface, the cylinder's curved surface and the cylinder's free base are exposed. The cone's base and cylinder's top base form the hidden circular contact and are excluded.

Example: For r = 3 cm, cone slant height l = 5 cm and cylinder height h = 8 cm, area = 15π + 48π + 9π = 72π cm².

Several Solids Joined Together

For more than two solids, add all exposed parts or subtract twice every internal contact area from the sum of the individual total surface areas.

If contact areas are C₁, C₂ and C₃, then external area = SA₁ + SA₂ + SA₃ + ... − 2(C₁ + C₂ + C₃ + ...). A contact area must be counted only when the two solids actually touch over that region.

Example: If three solids have total surface areas 40 cm², 50 cm² and 60 cm², with two separate contact areas of 6 cm² and 8 cm², external area = 150 − 2(6 + 8) = 122 cm².

Surface Area of Combined Solids Visual Guide

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

Cylinder joined to a hemisphere

Visual guide: Cylinder joined to a hemisphere
Common radius: r = 3 cm Cylinder height: h = 8 cm Exposed area: 2πrh + πr² + 2πr²
The circular junction is hidden; the cylinder's curved surface, bottom base and hemisphere's curved surface are exposed.

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Practice Surface Area of Combined Solids 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

Surface Area of Combined Solids: Revision Points

Use these rules to revise the calculation of external surface area.

  • Count only the outer exposed surfaces.
  • Exclude every surface at a junction between two solids.
  • If complete surface areas are added, subtract twice each common contact area.
  • Cylinder curved surface area = 2πrh.
  • Cone curved surface area = πrl.
  • Sphere surface area = 4πr².
  • Hemisphere curved surface area = 2πr² and total surface area = 3πr².
  • The final unit is always a square unit.

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 Combined Solids FAQs

What is the formula for the surface area of combined solids?

External surface area = sum of the individual total surface areas − 2 × total common contact area. Alternatively, add only the exposed surfaces directly.

Why is the common surface subtracted twice?

When individual total surface areas are added, the same contact surface is counted once for each joined solid. Therefore, it must be subtracted twice.

What is the surface area of a cylinder joined to a hemisphere?

For common radius r and cylinder height h, external surface area = 2πrh + πr² + 2πr² = 2πrh + 3πr².

What is the external area when r = 3 cm and h = 8 cm for a cylinder and hemisphere?

Area = 2π(3)(8) + π(3²) + 2π(3²) = 75π cm², approximately 235.62 cm².

Is the circular base of a hemisphere included when it is attached to a cylinder?

No. The hemisphere's circular base is the contact surface and becomes internal. Only its curved surface, 2πr², is included.

What is the external surface area of a cone joined to a cylinder?

If both have radius r, cone slant height l and cylinder height h, the external area is πrl + 2πrh + πr². The joined circular bases are excluded.

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