12.1-Surface Area of a Combination of Solids

12.1-Surface Area of a Combination of Solids Important Formulae

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Grade 10 → Math → Surface Areas and Volumes → 12.1-Surface Area of a Combination of Solids

After successful completion of this topic, you should be able to:

  • Apply formulas of surface area of 3D solids in order to derive the area of a new solid.

The surface area of a combination of solids involves finding the total surface area when two or more solids are combined. This is essential in various applications, such as architecture and manufacturing, where accurate measurements are crucial.

To calculate the surface area of combined solids, follow these steps:

  1. Identify the solids: Determine the types of solids involved in the combination, such as cubes, cylinders, cones, spheres, etc.
  2. Find the surface area of individual solids: Calculate the surface area of each solid using their respective formulas.

Common formulas for surface areas include:

  • Cube: Surface Area $= 6a^2$, where $a$ is the length of a side.
  • Cylinder: Total Surface Area $= 2\pi r(h + r)$, where $r$ is the radius and $h$ is the height.
  • Cone: Total Surface Area $= \pi r(l + r)$, where $l$ is the slant height and $r$ is the radius.
  • Sphere: Surface Area $= 4\pi r^2$, where $r$ is the radius.
  1. Determine shared surfaces: When solids are combined, certain surfaces may not contribute to the total surface area (e.g., where one solid is placed on top of another). Identify these shared surfaces to avoid double-counting.

For example, when a cylinder is placed on top of a cone, the circular base of the cone is not part of the total surface area. Thus, we need to subtract this area from the combined total.

  1. Calculate the total surface area: Combine the surface areas of the individual solids, making sure to subtract the areas of any shared surfaces.

Let's consider a practical example:

Suppose we have a cone with a base radius of $r = 3 \, \text{cm}$ and height $h = 4 \, \text{cm}$, placed on top of a cylinder with the same base radius and height $H = 5 \, \text{cm}$. We will find the total surface area of this combination.

First, calculate the surface areas:

  • Cone:
    • Calculate the slant height $l$ of the cone using the Pythagorean theorem:

      • $l = \sqrt{r^2 + h^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5 \, \text{cm}$

    • Surface Area of Cone $= \pi r(l + r) = \pi \cdot 3 \cdot (5 + 3) = 24\pi \, \text{cm}^2$
  • Cylinder:
    • Surface Area of Cylinder $= 2\pi r(H + r) = 2\pi \cdot 3 \cdot (5 + 3) = 48\pi \, \text{cm}^2$

Now, we add the surface areas of the cone and cylinder, subtracting the area of the circular base of the cone:

Area of the circular base of the cone $= \pi r^2 = \pi \cdot 3^2 = 9\pi \, \text{cm}^2$

Total Surface Area $= \text{Surface Area of Cylinder} + \text{Surface Area of Cone} - \text{Area of the base}$

Total Surface Area $= 48\pi + 24\pi - 9\pi = 63\pi \, \text{cm}^2$

This process of identifying individual surface areas and shared surfaces is crucial for accurately calculating the total surface area of combined solids.


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2 cubes each of volume 64 cm$^3$ are joined end to end. Find the surface area of the resulting cuboid.

Solution:

Surface Area of the Resulting Cuboid

Given the volume of each cube is 64 cm3, the side length of each cube can be calculated as:

Side length = ∛(Volume) = ∛(64) = 4 cm

When two cubes are joined end to end, the dimensions of the resulting cuboid will be:

  • Length = 4 cm + 4 cm = 8 cm
  • Width = 4 cm
  • Height = 4 cm

The surface area (SA) of a cuboid is given by the formula:

SA = 2(lw + lh + wh)

Substituting the values:

  • l = 8 cm
  • w = 4 cm
  • h = 4 cm

SA = 2(8*4 + 8*4 + 4*4) = 2(32 + 32 + 16) = 2(80) = 160 cm2

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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.

Solution:

Given:

  • Diameter of the hemisphere = 14 cm, so the radius $r = \frac{14}{2} = 7$ cm.
  • Total height of the vessel = 13 cm. The height of the hemisphere is equal to its radius, so the height of the cylinder = $13 - 7 = 6$ cm.

To find the inner surface area of the vessel, we need to calculate the surface area of the hemisphere and the lateral surface area of the cylinder.

1. Surface area of the hemisphere:

The surface area of a hemisphere (excluding the base) is given by $2\pi r^2$. Substituting $r = 7$ cm:

Surface area of hemisphere = $2\pi (7)^2 = 2\pi \times 49 = 98\pi$ cm2.

2. Lateral surface area of the cylinder:

The lateral surface area of a cylinder is given by $2\pi r h$, where $r$ is the radius and $h$ is the height. Substituting $r = 7$ cm and $h = 6$ cm:

Lateral surface area of the cylinder = $2\pi \times 7 \times 6 = 84\pi$ cm2.

Therefore, the inner surface area of the vessel is the sum of the surface area of the hemisphere and the lateral surface area of the cylinder:

Inner surface area = $98\pi + 84\pi = 182\pi$ cm2.

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A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy.

Solution:

Total Surface Area of the Toy

Given the radius (r) of the cone and hemisphere is 3.5 cm, and the total height of the toy is 15.5 cm.

The height of the cone (h) can be found as follows:

Height of the hemisphere = r = 3.5 cm

Height of the cone = Total height - Height of the hemisphere = 15.5 cm - 3.5 cm = 12 cm

Slant height of the cone = $\sqrt{12^2 + (3.5)^2}$= 12.5 cm

The total surface area (TSA) of the toy is the sum of the curved surface area of the cone and the curved surface area of the hemisphere:

Curved Surface Area of the cone = $\pi$rL = $\pi$(3.5)(12.5)

Curved Surface Area of the hemisphere = 2$\pi$r2 = 2$\pi$(3.5)2

Therefore, TSA = Curved Surface Area of the cone + Curved Surface Area of the hemisphere

TSA = $\pi$(3.5)(12.5) + 2$\pi$(3.5)2

TSA = $\pi$(3.5)(12.5 + 2(3.5))

TSA = $\pi$(3.5)(19.5)

TSA = $\dfrac{22}{7}$(3.5)(19.5)

TSA = $22(0.5)(19.5)$

TSA = $(11)(19.5)$

TSA = 214.5 cm$^2$

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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.

Solution:

Surface Area of the Solid

Given the side length of the cubical block is 7 cm, the greatest diameter of the hemisphere that can be surmounted on the cube is equal to the side of the cube.

Diameter of the hemisphere = 7 cm

Radius of the hemisphere (r) = Diameter / 2 = 7 cm / 2 = 3.5 cm

The total surface area (TSA) of the solid consists of the surface area of the cube and the curved surface area of the hemisphere.

Surface Area of the cube = 6a2 = 6(7)2 = 6(49) = 294 cm2

Curved Surface Area of the hemisphere = 2πr2 = 2π(3.5)2 = 2π(12.25) = 24.5π cm2

However, we need to subtract the area of the circular base of the hemisphere that is in contact with the cube.

Area of the circular base = πr2 = π(3.5)2 = π(12.25) cm2

Thus, the total surface area (TSA) is given by:

TSA = Surface Area of the cube + Curved Surface Area of the hemisphere - Area of the circular base

TSA = 294 + 24.5π - 12.25π

TSA = 294 + 12.25π cm2

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A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter l of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.

Solution:

Surface Area of the Remaining Solid

Let the edge of the cube be 'a'. The diameter of the hemisphere is equal to the edge of the cube, so:

Diameter of the hemisphere = a

Radius of the hemisphere (r) = a / 2

The surface area of the cube is given by:

Surface Area of the cube = 6a2

When a hemispherical depression is cut out, the curved surface area of the hemisphere is added, but the area of the circular base of the hemisphere is subtracted.

Curved Surface Area of the hemisphere = 2πr2 = 2π(a/2)2 = 2π(a2/4) = (πa2)

Area of the circular base = πr2 = π(a/2)2 = π(a2/4)

The surface area of the remaining solid is given by:

Surface Area = Surface Area of the cube + Curved Surface Area of the hemisphere - Area of the circular base

Surface Area = 6a2 + πa2 - π(a2/4)

Surface Area = 6a2 + πa2 - (πa2/4)

Surface Area = 6a2 + (4πa2/4) - (πa2/4)

Surface Area = 6a2 + (3πa2/4)

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A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends (see Fig. 12.10). The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area.

Solution:

Surface Area of the Medicine Capsule

Given the length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm.

Radius (r) of the capsule = Diameter / 2 = 5 mm / 2 = 2.5 mm

The length of the cylindrical part (h) can be calculated as:

Length of the cylindrical part = Total length - Length of two hemispheres

Length of two hemispheres = Diameter = 5 mm

Length of the cylindrical part (h) = 14 mm - 5 mm = 9 mm

The total surface area (TSA) of the capsule consists of the curved surface area of the cylinder and the surface area of the two hemispheres.

Curved Surface Area of the cylinder = 2πrh = 2π(2.5)(9)

Surface Area of the two hemispheres = 2 * (2πr2) = 2 * (2π(2.5)2)

Therefore, the total surface area is given by:

TSA = Curved Surface Area of the cylinder + Surface Area of the two hemispheres

TSA = 2π(2.5)(9) + 2 * (2π(2.5)2)

TSA = 2π(2.5)(9) + 4π(2.5)2

TSA = 2π(22.5) + 4π(6.25)

TSA = 45π + 25π = 70π mm2

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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. Also, find the cost of the canvas of the tent at the rate of Rs. 500 per m$^2$. (Note that the base of the tent will not be covered with canvas.)

Solution:

Area of the Canvas Used for Making the Tent

Given the height (hc) of the cylindrical part is 2.1 m and the diameter (d) is 4 m.

Radius (r) of the cylindrical part = d / 2 = 4 m / 2 = 2 m.

The slant height (l) of the conical top is 2.8 m.

The area of the canvas used for the tent consists of the curved surface area of the cylinder and the curved surface area of the cone.

Curved Surface Area of the cylinder = 2πrhc = 2π(2)(2.1).

Curved Surface Area of the cone = πrl = π(2)(2.8).

Thus, the total area of the canvas (A) is given by:

A = Curved Surface Area of the cylinder + Curved Surface Area of the cone

A = 2π(2)(2.1) + π(2)(2.8)

A = 8.4π + 5.6π = 14π m2

Cost of the canvas at the rate of Rs. 500 per m2:

Cost = Area × Rate = 14π × 500

Cost = 7000π Rs.

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From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid to the nearest cm$^2$.

Solution:

Total Surface Area of the Remaining Solid

Given the height (h) of the cylinder is 2.4 cm and the diameter (d) is 1.4 cm.

Radius (r) of the cylinder = d / 2 = 1.4 cm / 2 = 0.7 cm.

The height of the conical cavity is the same as that of the cylinder, h = 2.4 cm.

To find the slant height (l) of the cone, we use the Pythagorean theorem:

l = √(r2 + h2) = √(0.72 + 2.42)

l = √(0.49 + 5.76) = √(6.25) = 2.5 cm.

The curved surface area (CSA) of the cylinder is given by:

CSA of the cylinder = 2πrh = 2π(0.7)(2.4).

The curved surface area (CSA) of the cone is given by:

CSA of the cone = πrl = π(0.7)(2.5).

The area of the circular base of the cone is:

Area of the base = πr2 = π(0.7)2.

The total surface area (TSA) of the remaining solid is given by:

TSA = CSA of the cylinder + CSA of the cone + Area of the base of the cylinder - Area of the base of the cone

TSA = 2π(0.7)(2.4) + π(0.7)(2.5) + π(0.7)2 - π(0.7)2

TSA = 2π(0.7)(2.4) + π(0.7)(2.5)

TSA = 3.36π + 1.75π = 5.11π cm2

To the nearest cm2, TSA = 5.11 * 3.14 ≈ 16 cm2.

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A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 12.11. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the total surface area of the article.

Solution:

Total Surface Area of the Wooden Article

Given the height (h) of the cylinder is 10 cm and the radius (r) of the base is 3.5 cm.

The total surface area (TSA) of the wooden article consists of the curved surface area of the cylinder and the curved surface area of the two hemispheres.

Curved Surface Area of the cylinder (CSAcylinder) is given by:

CSAcylinder = 2πrh = 2π(3.5)(10).

Curved Surface Area of one hemisphere (CSAhemisphere) is given by:

CSAhemisphere = 2πr2 = 2π(3.5)2.

The total curved surface area of the two hemispheres is:

Total CSA of hemispheres = 2 * CSAhemisphere = 2 * (2π(3.5)2).

Thus, the total surface area of the article (TSA) is:

TSA = CSAcylinder + Total CSA of hemispheres

TSA = 2π(3.5)(10) + 2 * (2π(3.5)2)

TSA = 70π + 2 * (2π(12.25))

TSA = 70π + 49π = 119π cm2

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