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Solids are bodies that occupy space, such as a cuboid, a cube, a cylinder, a cone, a sphere, and so on. The surfaces of these solids are either flat or curved. The surface area of any three-dimensional object can be divided into three categories: Curved Surface Area (CSA), Lateral Surface Area (LSA), and Total Surface Area (TSA) (TSA). For 3d shapes such as a cube, cuboid, cone, cylinder, and so on, can be computed.
The surface area of a solid:
A 3D object’s surface area is the entire area covered by all of its faces. For example, if we need to calculate the amount of paint needed to paint a cube, the surface area is the area on which the paint will be applied. It is always expressed in terms of square units.
Volume and capacity of a solid:
The volume of a thing is the amount of space it takes up, while its capacity is the amount of substance it can hold in its interior. A cubic unit is a measuring unit for both volume and capacity.
Cuboid:
A three-dimensional Shape is a cuboid. The cuboid is formed by six rectangular faces placed at right angles. The sum of the areas of a cuboid’s six rectangular sides equals its entire surface area.
Total surface area of a cuboid:
Consider a cuboid with a length of “l” c., a width of “b” cm and a height of “h” cm.
Area of face ABCD = Area of Face EFGH = (l × b) cm²
Area of face AEHD = Area of face BFGC = (b × h) cm²
Area of face ABFE = Area of face DHGC = (l × h) cm²
TSA of cuboid = Sum of the areas of all its six faces
= 2(l × b) + 2(b × h) + 2(l × h)
TSA (cuboid)= 2(lb + bh +lh)
Volume of a cuboid:
A cuboid’s volume is equal to its dimensions multiplied by itself.
The Volume of a cuboid = length × breadth × height= lbh
Where l is the length of the cuboid, b is the breadth, and h is the height of the cuboid.
Cube:
A cube is a cuboid with the same length, width, and height on all sides. It’s a three-dimensional shape with six equal squares on all sides. It consists of 8 vertices and 12 edges.
Total surface area of a cube:
For cube, length = breadth = height
Assume that the length of an edge is equal to a.
Total surface area(TSA) of the cube = 2(a × a + a × a + a × a)
TSA(cube) = 2 × (3a²) = 6a²
The volume of a cube:
The volume of a cube = base area × height.
Because all dimensions are the same,
The volume of the cube = a³
The length of the cube’s edge is denoted by a.
Right circular cylinder:
A closed solid with two parallel circular bases connected by a curved surface in which the two bases are perfectly over each other and the axis is at right angles to the base is known as a right circular cylinder.
The total surface area of a right circular cylinder:
TSA of a cylinder of base radius r and height h = 2π × r × h + area of two circular bases
⇒ TSA = 2π × r × h + 2 × πr²
⇒ TSA = 2πr(h + r)
The volume of a right circular cylinder:
A right circular cylinder’s volume is equal to its base area multiplied by its height.
The volume of the cylinder =πr²h
Where r is the radius of the cylinder’s base and h is the cylinder’s height.
Right circular cone:
A right circular cone is one with an axis that is perpendicular to its base.
Total surface area of right circular cone:
Total surface area(TSA) = Curved surface area(CSA) + area of base
= πrl + πr²
= πr(l + r)
The volume of a right circular cone:
A right circular cone has a volume that is 1/3 that of a cylinder with the same radius and height. To put it another way, three cones form a cylinder with the same height and base.
The volume of right circular cone = (1/3)πr²h
The radius of the cone’s base is r, while the height of the cone is h.
Conclusion:
We employ the concept of surface areas of different items in real life when we want to wrap something, paint something, and eventually build something to obtain the greatest design possible. Determining the volume of an object can be quite useful. Knowing the volume of a thing can help you figure out how much you’ll need to fill it, such as how much water you’ll need to fill an aquarium.
