Surface Area of Combined Solids

When two or more solid shapes are joined together, the result is called a combined solid. To find its surface area, you add up all the outer surfaces that are visible from the outside. Any surface that is hidden where the two shapes meet must be removed, since it is no longer part of the outer covering.

Table of Contents

• What is the Surface Area of Combined Solids?
• Surface Area of a Combination of Solids Formula
• Surface Area of a Combination of Solids Examples
• Surface Area of a Combination of Solids Practice Problems

What is the Surface Area of Combined Solids?

Surface area is the total outer area of a solid shape. When two shapes are joined, one part of each shape gets covered at the joint. You must remove those hidden parts and add only what remains exposed on the outside.

Total surface area = sum of all outer surfaces − the hidden (joined) surfaces.

The red dashed line above shows the joint between a cone and a cylinder. That circular edge is inside the shape it does not appear on the outer surface, so it is not included in the surface area.

Surface Area of a Combination of Solids Formula

There is no single formula for every combined solid because the answer depends on which shapes are joined. However, the rule is always the same:

Surface area of combined solid = (surface area of shape A + surface area of shape B) − 2 × (area of the joined face)

You subtract the joined face twice once from each shape because both shapes lose that face at the point of contact.

Here are the most common combinations and how the rule applies to each:

1. Cylinder + Cone:

The cone sits on top of the cylinder. The flat circular base of the cone and the top circle of the cylinder are both hidden at the joint.

Surfaces that count: Curved surface of cone + curved surface of cylinder + base circle of cylinder

Surface that does NOT count: The circular face where the cone and cylinder meet (hidden inside)

2. Cylinder + Hemisphere:

The hemisphere (half-sphere) sits on top of the cylinder. The flat circular face of the hemisphere and the top circle of the cylinder disappear at the joint.

Surfaces that count: Curved surface of hemisphere + curved surface of cylinder + base circle of cylinder

Surface that does NOT count: The flat circular face of the hemisphere where it sits on the cylinder

3. Cone + Hemisphere:

The cone sits on top of the flat side of the hemisphere. Both shapes share a circular face at the joint which becomes hidden.

Surfaces that count: Curved surface of cone + curved surface of hemisphere

Surface that does NOT count: The flat circular face shared by both shapes at the joint. There is no flat base here at all the hemisphere itself forms the rounded bottom.

Surface Area of a Combination of Solids Examples

Example 1: Cylinder + Cone (like a rocket or tent)

A cylinder of radius 7 cm and height 10 cm has a cone of the same radius and slant height 13 cm placed on top.

Step 1: Identify the visible surfaces:

• Curved surface of the cone
• Curved surface of the cylinder
• Base circle of the cylinder (bottom flat face)

Step 2: The hidden surface:

• The top circle of the cylinder and the base of the cone are the same circle they touch at the joint and are both hidden.

Step 3: Apply the values (r = 7, h = 10, l = 13):

• Curved surface of cone = π × 7 × 13 = 286 cm²
• Curved surface of cylinder = 2 × π × 7 × 10 = 440 cm²
• Base of cylinder = π × 7² = 154 cm²

Total surface area ≈ 880 cm²

Example 2: Cylinder + Hemisphere (like a water tank)

A cylinder of radius 3.5 m and height 8 m has a hemispherical dome of the same radius on top.

Step 1: Identify the visible surfaces:

• Curved surface of the hemisphere
• Curved surface of the cylinder
• Base circle of the cylinder

Step 2: The hidden surface:

• The flat face of the hemisphere sits on the top circle of the cylinder both are hidden at the joint.

Step 3: Apply the values (r = 3.5, h = 8):

• Curved surface of hemisphere = 2 × π × 3.5² = 77 m²
• Curved surface of cylinder = 2 × π × 3.5 × 8 = 176 m²
• Base circle = π × 3.5² = 38.5 m²

Total surface area ≈ 291.5 m²

Example 3: Cone + Hemisphere (like a spinning top)

A hemisphere of radius 6 cm has a cone of the same radius and slant height 10 cm placed on top.

Step 1: Identify the visible surfaces:

• Curved surface of the cone
• Curved surface of the hemisphere

Step 2: The hidden surface:

• The flat face where the cone base meets the hemisphere is hidden. There is no flat base circle here at all.

Step 3: Apply the values (r = 6, l = 10):

• Curved surface of cone = π × 6 × 10 = 188.5 cm²
• Curved surface of hemisphere = 2 × π × 6² = 226.2 cm²

Total surface area = 414.7 cm²

Surface Area of a Combination of Solids Practice Problems

1. A cylinder of radius 5 cm and height 12 cm has a hemisphere attached on top. Find the total surface area of the solid?

2. A cone is placed on a cylinder. Both have the same base radius of 3 cm. The height of the cylinder is 8 cm, and the slant height of the cone is 5 cm. Calculate the total surface area?

3. A solid is formed by joining two hemispheres to the ends of a cylinder. The radius is 4 cm, and the height of the cylinder is 10 cm. Find the total surface area?

4. A cube of side 10 cm has a hemisphere carved out from one face. Find the surface area of the remaining solid?

5. A cylinder is surmounted by a cone of the same radius, 6 cm. The height of the cylinder is 15 cm, and the slant height of the cone is 10 cm. Find the total surface area of the combined solid?