How do I use these formulas for points on a cube or cuboid?

In one line

The rule

The volume of a cube is the space inside it, found by multiplying the length of one side by itself three times. The volume of a cuboid is found by multiplying its length, width, and height. The distance between two points on a cube or cuboid can be found using the Pythagoras theorem (for straight lines).

Formulas:

• Volume of cube: V = side³ (side = length of one edge)
• Volume of cuboid: V = l × w × h (l = length, w = width, h = height)
• Cube root: ∛x (the number which, when multiplied by itself three times, gives x)

Let's understand it simply

Imagine you have a dice (which is a cube) or a matchbox (which is a cuboid). If you want to know how much space is inside, you use the volume formula. If you want to find out how far it is from one corner to another, you use the distance formula. Think of the corners as points. Formulas help you measure things between these points.

How it works

1. To find the volume, measure the sides (cube) or length, width, and height (cuboid).
2. Multiply them as per the formula.
3. To find the distance between two points (for example, from one corner to the opposite corner), use the Pythagoras theorem:

• For a cube with side a, the space diagonal (corner to farthest corner) is a√3.
• For a cuboid with length l, width w, height h, the space diagonal is √(l² + w² + h²).

1. If you know the volume and want to find the side of a cube, use the cube root: side = ∛(volume).

The story behind it

The formulas for cubes and cuboids have been used for thousands of years, even by ancient builders. The Pythagoras theorem, used for finding distances, was named after the Greek mathematician Pythagoras, who lived around 2,500 years ago. Builders needed these formulas to measure rooms, boxes, and storage spaces.

Where you see it in real life

• Dice in Ludo: Each corner is a point; the distance between corners can be found using these formulas.
• A water tank (cuboid): To know how much water it holds, use the volume formula.
• A classroom: To find how much air is inside, measure length, width, and height.
• A Diwali gift box: To check if a toy fits inside, compare the size of the toy to the box's inside space.

Why it matters

• Helps in packing and storage: Know if things will fit.
• Used in construction: Builders need to know how much material is needed.
• Important for exams: Many geometry questions use these formulas.
• Used in science: To find the volume of objects for experiments.

Worked example

Let's say you have a cuboid with length = 4 cm, width = 3 cm, height = 2 cm. Find the distance from one corner to the opposite corner (space diagonal).

Given: l = 4 cm, w = 3 cm, h = 2 cm
Step 1: Write the formula for space diagonal: d = √(l² + w² + h²)
Step 2: Substitute values: d = √(4² + 3² + 2²)
Step 3: Calculate squares: d = √(16 + 9 + 4)
Step 4: Add: d = √29
Step 5: Find the square root (approximate): d ≈ 5.39 cm

Common mistakes

• Mixing up length, width, and height: Always check which side is which.
• Forgetting to use the cube root when finding the side from volume.
• Using the wrong formula for diagonal (using only two sides instead of all three for space diagonal).
• Not using the same units (cm, m, etc.) for all measurements.

Remember it

To measure space or distance in a cube or cuboid, use the right formula—volume for inside, Pythagoras for corners.