**Three-dimensional geometry** is a branch of mathematics that deals with the properties and relationships of objects in three-dimensional space. It is an essential part of many fields such as physics, engineering, and computer graphics. Understanding the concepts of three-dimensional geometry is crucial for solving problems related to distance, volume, and surface area.

One of the fundamental concepts in three-dimensional geometry is the** distance** formula. This formula is used to find the distance between any two points in three-dimensional space.

The **formula** is given by:

**Distance** = √((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)

For example, if two points in three-dimensional space are given as (3, 5, 8) and (6, 8, 12), the distance between the two points can be found by substituting these values into the

**formula:**

**Distance** = √((6-3)^2 + (8-5)^2 + (12-8)^2) = √(3^2 + 3^2 + 4^2) = √9 + √9 + √16 = √34

Another important concept in three-dimensional geometry is volume. The **volume** of an object in three-dimensional space is a measure of the amount of space it occupies.

**The formula for the volume of a rectangular prism is given by:**

**Volume** = Length x Width x Height

For example, if a rectangular prism has a length of 5 cm, a width of 3 cm, and a height of 2 cm, the volume of the rectangular prism can be found by substituting these values into

__the formula:__

**Volume** = 5 x 3 x 2 = 30 cm^3

**Surface area** is another important concept in three-dimensional geometry. The surface area of an object in three-dimensional space is a measure of the total area of its surfaces.

The formula for the surface area of a rectangular prism is given by:

**Surface area** = 2(lw + lh + wh)

For example, if a rectangular prism has a length of 5 cm, a width of 3 cm, and a height of 2 cm, the surface area of the rectangular prism can be found by substituting these values into the formula:

**Surface area** = 2(5 x 3 + 5 x 2 + 3 x 2) = 2(15 + 10 + 6) = 2(31) = 62 cm^2

In conclusion, three-dimensional geometry is an essential branch of mathematics that deals with the properties and relationships of objects in three-dimensional space. Understanding the concepts of distance, volume, and surface area is crucial for solving problems related to three-dimensional geometry. With practice and persistence, anyone can master the concepts and solve problems with ease.

Practice Questions on 3D Geometry:

1. **What is the formula for the distance between two points in three-dimensional space**?

a) Distance = √((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)

b) Distance = (x2-x1) + (y2-y1) + (z2-z1)

c) Distance = (x2-x1) x (y2-y1) x (z2-z1)

d) Distance = √((x2-x1) + (y2-y1) + (z2-z1))

**2. What is the formula for the volume of a rectangular prism? **

a) Volume = Length x Width x Height

b) Volume = (Length + Width + Height) / 3

c) Volume = (Length x Width) / 2

d) Volume = (Length + Width) x Height

**3. What is the formula for the surface area of a rectangular prism?**

a) Surface area = (Length x Width) + (Length x Height) + (Width x Height)

b) Surface area = 2(lw + lh + wh)

c) Surface area = (Length x Width x Height) / 2

d) Surface area = (Length + Width + Height) x 2

**4. A cube has a side length of 5cm, what is the volume of the cube?**

a) 25 cm^3

b) 15 cm^3

c) 20 cm^3

d) 30 cm^3

**5. A cylinder has a radius of 3cm and a height of 6cm, what is the surface area of the cylinder? **

a) 84 cm^2

b) 94 cm^2

c) 74 cm^2

d) 64 cm^2

**6. A sphere has a radius of 4cm, what is the volume of the sphere? **

a) 64π cm^3

b) 16π cm^3

c) 32π cm^3

d) 48π cm^3

Answers:

a

a

b

a

a

c

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