Surface Area and Volume

Imagine you have been given a brand new box, a wonderful gift filled with all kinds of sweets. You are curious to know how much cardboard was used to make the box or how many sweets can be stored in the box. These questions refer to two concepts of mathematics: surface area and volume, which are part of the field of study known as measurement.

What is the surface area?

Surface area is the total area that a three-dimensional object covers. If you cut your gift box open and lay it flat, the amount of cardboard you have laid out is the surface area.

The surface area is different for different shapes. Let's learn about these different shapes and how to find their surface area.

Cuboid

A cuboid has six faces: top, bottom, left side, right side, front, and back. Each face is a rectangle, and the opposite faces are identical. To calculate the surface area, you need to find the area of each face and then add these areas together.

To understand this better, let's take a rectangle with length ( l ), width ( b ) and height ( h ).

Surface area of cuboid = 2(lb + bh + hl)

Cube

A cube is a special type of cuboid where the length, width, and height are all equal. This simplifies calculating surface area because each of the six faces of a cube is a square of side a.

Surface area of the cube = 6a 2

Right circular cylinder

A cylinder has one curved surface and two flat circular bases; so its surface area includes the area of both the circles and the curved surface.

Curved surface area of cylinder = 2πrh

Area of both circular bases = 2πr 2

Total surface area of cylinder = 2πrh + 2πr² = 2πr(h + r)

Circle

A sphere is a perfectly round 3D shape, where every point on its surface is the same distance from its center. Calculating the surface area of a sphere is simple because it has no extra faces or edges.

Surface area of sphere = 4πr²

What is volume?

Volume is the space occupied by a 3D object. Consider the sweets box again, the volume would be the internal capacity of the box, which tells you how many sweets it can hold.

Like surface area, calculating volume also varies with different shapes.

Volume of a cuboid

The volume of a cuboid is found by multiplying its length, breadth, and height.

Volume of cuboid = l × b × h

Volume of a cube

Given that a cube has all equal sides, its volume is simply the cube of the side length.

Volume of the cube = a 3

Volume of a right circular cylinder

The volume of a cylinder is determined by multiplying the area of its circular base by its height.

Volume of cylinder = πr²h

Volume of a sphere

The volume of a sphere is calculated using the radius of the sphere, and uses π formula.

Volume of sphere = (4/3)πr³

Examples of calculating surface area and volume

Example 1: Cuboid

A cuboid has a length of 5 cm, width of 3 cm and height of 2 cm. What is its surface area and volume?

Calculate the surface area:

Surface area = 2(lb + bh + hl)
= 2(5 × 3 + 3 × 2 + 2 × 5)
= 2(15 + 6 + 10)
= 2 × 31
= 62 cm²

Calculate the volume:

Volume = L × W × H
= 5 × 3 × 2
= 30 cm³

Example 2: Cube

The length of the side of a cube is 4 cm. What is its surface area and volume?

Calculate the surface area:

Surface area = 6a²
= 6 × 4²
= 6 × 16
= 96 sq.cm

Calculate the volume:

Volume = a³
= 4³
= 64 cm³

Example 3: Right circular cylinder

A cylinder has a radius of 3 cm and a height of 7 cm. What is its surface area and volume?

Calculate the surface area:

Surface area = 2πr(h + r)
= 2 × 3.14 × 3 × (7 + 3)
= 18.84 × 10
= 188.4 cm²

Calculate the volume:

Volume = πr²h
= 3.14 × 3² × 7
= 3.14 × 9 × 7
= 3.14 × 63
= 197.82 cm³

Example 4: Sphere

The radius of a sphere is 5 cm. What is its surface area and volume?

Calculate the surface area:

Surface area = 4πr²
= 4 × 3.14 × 5²
= 4 × 3.14 × 25
= 314 sq. cm

Calculate the volume:

Volume = (4/3)πr³
= (4/3) × 3.14 × 5³
= (4/3) × 3.14 × 125
= 523.33 cm³

Conclusion

In mathematics, understanding surface area and volume helps us understand the extent and capacity of three-dimensional objects. Surface area helps us know how much material is needed to cover an object, while volume tells us how much space it takes up. Appreciating these concepts builds foundational skills for higher-level mathematics and provides practical utility in many real-world applications.

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