Surface Areas of Combination of Solids (NCERT Ex 13.1)

The Core Concept: Visualizing Combined Surface Area

When we combine two or more solids, a common mistake is to simply add their individual Total Surface Areas (TSA). This is incorrect! When you join two solids, the faces where they are joined get covered and are no longer part of the 'surface'. For example, if you place a hemisphere on top of a cylinder, the circular base of the hemisphere and the circular top of the cylinder are no longer exposed. Therefore, they are not included in the total surface area of the new, combined shape. The correct method is to find the sum of the Curved Surface Areas (CSA) of the visible parts. For the hemisphere on a cylinder, you would calculate the CSA of the hemisphere and the CSA of the cylinder. If the cylinder also has a base at the bottom, you would add the area of that circular base. The formula is: Total Surface Area of Combined Solid = Sum of Curved Surface Areas of all visible parts + Area of any visible flat surfaces.

Revision: Key Surface Area Formulas

Cube

Let side be 'a'. TSA = 6a², CSA = 4a²

Cuboid

Let dimensions be l, b, h. TSA = 2(lb + bh + hl), CSA = 2(l+b)h

Cylinder

Let radius be 'r' and height be 'h'. TSA = 2πr(r + h), CSA = 2πrh

Cone

Let radius be 'r', height 'h', slant height 'l'. TSA = πr(l + r), CSA = πrl. Remember, l = √(r² + h²).

Sphere

Let radius be 'r'. TSA = 4πr²

Hemisphere

Let radius be 'r'. TSA = 3πr², CSA = 2πr²

A Step-by-Step Guide to Solving Problems

1. Step 1: Visualize and Sketch — Read the problem carefully and draw a rough sketch of the combined solid. This is the most important step. Label all the given dimensions like height, radius, and side lengths.
2. Step 2: Identify Component Solids — Break down the composite shape into its basic solid components (e.g., a cone, a cylinder, a hemisphere).
3. Step 3: Determine the Visible Surfaces — Identify which parts of each component solid are visible on the outside. Remember that the surfaces where the solids are joined are NOT visible and should not be included.
4. Step 4: Apply Formulas and Calculate — Choose the correct formula (usually CSA) for each visible part. Calculate the area for each part separately. Be careful with units.
5. Step 5: Sum the Areas — Add the areas calculated in the previous step to get the total surface area of the combined solid. Write the final answer with the correct units (e.g., cm², m²).

Worked Example: Toy as a Cone on a Hemisphere

• Problem: A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of the same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy. (Use π = 22/7) Solution: Step 1: Identify dimensions Radius of cone (r) = Radius of hemisphere (r) = 3.5 cm. Total height of toy = 15.5 cm. Height of hemisphere = its radius = 3.5 cm. Height of cone (h) = Total height - Height of hemisphere = 15.5 - 3.5 = 12 cm. Step 2: Find the slant height of the cone (l) The formula is l = √(r² + h²). l = √((3.5)² + (12)²) l = √(12.25 + 144) l = √156.25 l = 12.5 cm. Step 3: Identify the visible surfaces The visible surfaces are the Curved Surface Area (CSA) of the cone and the Curved Surface Area (CSA) of the hemisphere. The circular base where they join is hidden. Step 4: Calculate the areas CSA of cone = πrl = (22/7) × 3.5 × 12.5 = 22 × 0.5 × 12.5 = 137.5 cm². CSA of hemisphere = 2πr² = 2 × (22/7) × (3.5)² = 2 × (22/7) × 12.25 = 2 × 22 × 1.75 = 77 cm². Step 5: Sum the areas Total Surface Area of the toy = CSA of cone + CSA of hemisphere Total Surface Area = 137.5 + 77 = 214.5 cm². Final Answer: The total surface area of the toy is 214.5 cm².

Exam Tip: Avoid This Common Mistake!

The most frequent error in these problems is mechanically adding the Total Surface Areas (TSA) of the individual solids. For a cone mounted on a hemisphere, students often calculate TSA of cone + TSA of hemisphere. This is wrong! The circular base of the cone and the circular top of the hemisphere are glued together and are internal, not external. Always visualize the final object. Think: 'If I were to paint this object, which surfaces would I paint?' You would only paint the curved part of the cone and the curved part of the hemisphere. Therefore, you must add their Curved Surface Areas (CSA), not their Total Surface Areas.

Practice Questions with Solutions

• Q: 2 cubes each of volume 64 cm³ are joined end to end. Find the surface area of the resulting cuboid. A: Step 1: Find the side of each cube. Volume of a cube = a³. So, a³ = 64 cm³. This gives the side, a = 4 cm. Step 2: Determine the dimensions of the new cuboid. When two cubes are joined end to end, the length becomes a + a = 4 + 4 = 8 cm. The breadth (b) and height (h) remain the same, so b = 4 cm and h = 4 cm. Step 3: Calculate the surface area of the cuboid. The formula is TSA = 2(lb + bh + hl). TSA = 2((8)(4) + (4)(4) + (4)(8)) TSA = 2(32 + 16 + 32) = 2(80) = 160 cm². Final answer: The surface area of the resulting cuboid is 160 cm².
• Q: A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel. A: Step 1: Find the radius and heights. Diameter = 14 cm, so radius (r) = 7 cm. This is the radius for both the hemisphere and the cylinder. Total height = 13 cm. Height of hemispherical part = radius = 7 cm. Height of cylindrical part (h) = Total height - Height of hemisphere = 13 - 7 = 6 cm. Step 2: Identify the inner surfaces. The inner surface consists of the CSA of the cylinder and the CSA of the hemisphere. Step 3: Calculate the areas. Inner CSA of cylinder = 2πrh = 2 × (22/7) × 7 × 6 = 264 cm². Inner CSA of hemisphere = 2πr² = 2 × (22/7) × 7² = 2 × 22 × 7 = 308 cm². Step 4: Add the areas. Total inner surface area = 264 + 308 = 572 cm². Final answer: The inner surface area of the vessel is 572 cm².
• Q: A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid. A: Step 1: Determine the greatest diameter. The hemisphere is on top of the cube. The greatest diameter it can have is equal to the side length of the cube. So, greatest diameter = 7 cm, which means radius (r) = 3.5 cm. Step 2: Identify the visible surfaces. The total surface area of the solid = TSA of the cube - Area of the base of the hemisphere + CSA of the hemisphere. The base of the hemisphere covers a part of the cube's top face, so we must subtract it. Step 3: Calculate the areas. TSA of cube = 6a² = 6 × 7² = 6 × 49 = 294 cm². Area of base of hemisphere = πr² = (22/7) × (3.5)² = 38.5 cm². CSA of hemisphere = 2πr² = 2 × (22/7) × (3.5)² = 77 cm². Step 4: Calculate the total surface area. Total Area = 294 - 38.5 + 77 = 255.5 + 77 = 332.5 cm². Final answer: The greatest diameter is 7 cm and the total surface area is 332.5 cm².
• Q: A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m, find the area of the canvas used for making the tent. (Note that the base of the tent will not be covered with canvas). A: Step 1: Identify dimensions. For the cylinder: diameter = 4 m, so radius (r) = 2 m. Height (h) = 2.1 m. For the cone: radius (r) = 2 m. Slant height (l) = 2.8 m. Step 2: Identify surfaces to be covered by canvas. The canvas covers the CSA of the cylindrical part and the CSA of the conical part. Step 3: Calculate the areas. CSA of cylinder = 2πrh = 2 × (22/7) × 2 × 2.1 = 2 × 22 × 2 × 0.3 = 26.4 m². CSA of cone = πrl = (22/7) × 2 × 2.8 = 22 × 2 × 0.4 = 17.6 m². Step 4: Add the areas. Total canvas area = CSA of cylinder + CSA of cone = 26.4 + 17.6 = 44 m². Final answer: The area of the canvas used is 44 m².

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