Formulas for Calculating Volume, Surface Area, and Diagonals of Basic Geometric Solids

This guide provides formulas for calculating the volume, surface area, and diagonals of common geometric solids. Let’s break down these important shapes and their key properties:

1. Pyramid

• Volume of a Pyramid: $V = frac{1}{3} . S_{base} . h$

• Where:

• $V$: Volume of the pyramid

• $S_{base}$: Area of the base of the pyramid

• $h$: Height of the pyramid

2. Regular Square Pyramid

• Area of an Equilateral Triangle: $S = frac{a^2 sqrt{3}}{4}$

• Where:

• $S$: Area of the equilateral triangle

• $a$: Side length of the equilateral triangle

• Volume of a Regular Square Pyramid: $V = frac{1}{3} . S_{base} . h = frac{1}{3} . a^2 . h$

• Where:

• $V$: Volume of the pyramid

• $a$: Side length of the square base

• $h$: Height of the pyramid

3. General Prism

• Volume of a Prism: $V = S_{base} . h$

• Where:

• $V$: Volume of the prism

• $S_{base}$: Area of the base of the prism

• $h$: Height of the prism

4. Cube

• Space Diagonal of a Cube: $d = a sqrt{3}$

• Where:

• $d$: Space diagonal (longest diagonal) of the cube

• $a$: Side length of the cube

• Volume of a Cube: $V = a^3$

• Where:

• $V$: Volume of the cube

• $a$: Side length of the cube

5. Square

• Diagonal of a Square: $d = a sqrt{2}$

• Where:

• $d$: Diagonal of the square

• $a$: Side length of the square

6. Cone

• Volume of a Cone: $V = frac{1}{3} . pi . r^2 . h$

• Where:

• $V$: Volume of the cone

• $r$: Radius of the base of the cone

• $h$: Height of the cone

• Lateral Surface Area of a Cone: $S_{lateral} = pi . r . l$

• Where:

• $S_{lateral}$: Lateral surface area (area of the curved side) of the cone

• $r$: Radius of the base of the cone

• $l$: Slant height of the cone (the distance from the apex to a point on the edge of the base)

• Total Surface Area of a Cone: $S_{total} = S_{lateral} + S_{base} = pi . r . l + pi . r^2$

• Where:

• $S_{total}$: Total surface area of the cone

• $S_{lateral}$: Lateral surface area of the cone

• $S_{base}$: Area of the base of the cone

• $r$: Radius of the base of the cone

• $l$: Slant height of the cone

7. Sphere

• Volume of a Sphere: $V = frac{4}{3} . pi . r^3$

• Where:

• $V$: Volume of the sphere

• $r$: Radius of the sphere

• Surface Area of a Sphere: $S = 4 . pi . r^2$

• Where:

• $S$: Surface area of the sphere

• $r$: Radius of the sphere

Important Notes:

• These formulas are for basic geometric solids. They may not apply to more complex shapes.

• In some cases, you might need to use additional formulas (e.g., for area, height, slant height) to calculate the volume or surface area of a solid.

• Pay close attention to the units of measurement when applying these formulas. Consistency is crucial!

Example:

• Calculate the volume of a triangular pyramid with an equilateral triangular base of side length 6 cm and a height of 4 cm.

• $S_{base} = frac{a^2 sqrt{3}}{4} = frac{6^2 sqrt{3}}{4} = 9sqrt{3}$ (cm^2)

• $V = frac{1}{3} . S_{base} . h = frac{1}{3} . 9sqrt{3} . 4 = 12sqrt{3}$ (cm^3)

• Calculate the surface area of a sphere with a radius of 5 cm.

• $S = 4 . pi . r^2 = 4 . pi . 5^2 = 100pi$ (cm^2)