In our daily lives, we come across various cones of different shapes and sizes for which we can calculate surface area. We know how to calculate the surface area of the items in our surroundings. But what if these basic forms combine to produce a new shape that isn’t the same as the original? 

The challenge now is figuring out how to determine the surface and area of new objects. We must notice the new form while calculating the surface area of these new forms. In this article, we will discuss the combination of cone and hemisphere and their surface area in detail.

Surface Area of a Combination of Cone and Hemisphere: Introduction

Have you ever come across a capsule? It consists of two hemispheres joined by a cylinder.

In everyday life, we’ll see shapes that are made up of different solids. These forms can be hollow or solid, and they can be studied as a combination of solids.

Volume and Surface Area of Solid

The surface area of a three-dimensional figure is the amount of exterior space that it covers. A solid shape’s surface area can be the lateral surface area, curved surface area, or total surface area.

A solid shape’s volume is defined as the amount of space it takes up. It is the space bounded by a border, occupied by an object, or capable of holding something.

Combination of Solids

Solid forms are three-dimensional structures that are otherwise planar shapes. When converted to a three-dimensional structure, a square becomes a cube, a rectangle becomes a cuboid, and a triangle becomes a cone. In the case of planar shapes, we can only measure the area of the shape.

But when we deal with \(3-D\) shapes, we intend to measure their volume, surface area or curved surface area. Solid shapes, as already said, are \(3-D\) counterparts of their planar shape. But what if these solid shapes join together to form a new shape? A different level of measurement is obtained by combining solids.

We observe many shapes that combine different shapes in our daily lives, like huts, tents, capsules, and ice-cream-filled cones. So, what precisely is a solid combination? The figure created by combining two or more different solids is known as a combination of solids.

Examples of Combinations of Solids

Check some of the examples of combinations of solids below

A Circus Tent or a Hut

A circus tent is a cylinder and a cone combination. A cuboid and a cone can also be seen in circus tents. A hut is a type of kutcha home that resembles a tent.

An Ice Cream Cone

A right circular cone is combined with a hemisphere with the same circular base as the cone to form an ice cream cone.

A Dome on a Solid Shape

A dome is usually constructed on top of a structure or a tent. The upper part of the hemisphere is known as a dome. Now, if a structure has a dome, we combine the solid shape of the structure with the dome shape. An apex dome, which is a combination of a hemisphere and a cylinder, is seen below.

Mushroom

Mushrooms have a cylinder-shaped body with a cone-shaped top.

Funnel

A funnel is a combination of a frustum of a cone and a cylinder.

Pencil

A sharpened pencil is a combination of a cone and a cylinder.

Surface Areas Formulas

The surface area of various solid shapes are given below:

1. Cuboid

Lateral Surface Area \( = 2\left( {l + b} \right)h\)

Total Surface Area \(=2(l b+b h+h l)\)

Where \(l, b\), and \(h\) are the length, breadth and height of a cuboid.

2. Cube

Lateral Surface Area \(=4 a^{2}\)

Total Surface Area \(=6 a^{2}\)

Where \(a\) is the side of the cube.

3. Cylinder

Curved Surface Area \(=2 \pi r h\)

Total Surface Area \(=2 \pi r(r+h), r\) is the radius of the circular base, and \(h\) is the height of the cylinder.

4. Cone

Curved Surface area \(=\pi r l\)

Total Surface Area \(=\pi r(l+r), r\) is the radius of the circular base, \(l\) is the slant height of the cone.

5. Sphere

Curved Surface Area \(=4 \pi r^{2}\)

Total Surface Area \(=4 \pi r^{2}\)

Where \(r\) is the radius of the sphere

6. Hemisphere

Curved Surface Area \(=2 \pi r^{2}\)

Total Surface Area \(=3 \pi r^{2}\)

Where \(r\) is the radius of the hemisphere.

7. Frustum of a Cone

Lateral/Curved Surface Area \(=\pi l\left(r_{1}+r_{2}\right)\)

Total Surface Area \(=\pi l\left(r_{1}+r_{2}\right)+\pi r_{1}^{2}+\pi r_{2}^{2}, r_{1}\) and \(r_{2}\) are the radius of the circular bases, \(l\) is the slant height.

Surface Area of Combination of Solids

When working with solid structure calculations, we need to be extra cautious with our measurements. Finding the surface area of a solid combination requires logic and expertise. In such computations, the initial step is to figure out what shapes have combined to form the structure.

Finding the surface area for a structure becomes simple and quick once you’ve figured out the basic shapes. You must combine the surface areas of the constituting structures to determine the surface area of a solid structure created by combining two or more solids.

To calculate the surface area of a circus tent, combine the cone and cylinder surface areas. In a tent made up of a cone and a cylinder, we first compute the surface area of each cone and cylinder separately, then combine them.

For example, consider three cubes each of \(5 \,\text {cm}\) edge are joined end to end. Find the surface area of the resulting cuboid.

If three cubes are joined end to end, we get a cuboid such that,

length of the resulting cuboid, \(l=5 \mathrm{~cm}+5 \mathrm{~cm}+5 \mathrm{~cm}=15 \mathrm{~cm}\)

the breadth of the resulting cuboid, \(b=5 \mathrm{~cm}\)

height of the resulting cuboid, \(h=5 \mathrm{~cm}\)

The surface area of the cuboid \(=2(l b+b h+l h)\)

\(=2(15 \times 5+5 \times 5+5 \times 15)\)

\(=2(75+25+75)=2(175)=350 \mathrm{~cm}^{2}\)

We hope that this article on the Magnetic Field and Field Lines has helped you. If you have any queries or doubts, feel to contact us by writing to us at [email protected]

Stay tuned to Embibe to learn more on such topics.