Comprehensive reference guide for calculating area, surface area, and volume of common geometric shapes with formulas and examples.

The image provided is a reference sheet for calculating Area, Surface Area, and Volume of various two-dimensional and three-dimensional shapes. Below, I will explain the concepts and formulas presented in the table.

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1. Two-Dimensional Shapes (Area)

#### Square
- Formula: \( \text{Area} = a^2 \) or \( a \times a \)
- Explanation: The area of a square is calculated by squaring the length of one of its sides.
- Example: If \( a = 5 \, \text{cm} \), then \( \text{Area} = 5^2 = 25 \, \text{cm}^2 \).

#### Rectangle
- Formula: \( \text{Area} = w \times h \)
- Explanation: The area of a rectangle is calculated by multiplying its width by its height.
- Example: If \( w = 10 \, \text{cm} \) and \( h = 20 \, \text{cm} \), then \( \text{Area} = 10 \times 20 = 200 \, \text{cm}^2 \).

#### Triangle
- Formula: \( \text{Area} = b \times h \times 0.5 \)
- Explanation: The area of a triangle is half the product of its base and height.
- Example: If \( b = 20 \, \text{cm} \) and \( h = 15 \, \text{cm} \), then \( \text{Area} = 20 \times 15 \times 0.5 = 150 \, \text{cm}^2 \).

#### Regular Polygon
- Formula: \( \text{Area} = n \times s \times a \times 0.5 \)
- Explanation: The area of a regular polygon is calculated by multiplying the number of sides (\( n \)), the length of each side (\( s \)), the apothem (\( a \)), and 0.5.
- Example: If \( n = 6 \), \( s = 5 \, \text{cm} \), and \( a = 15 \, \text{cm} \), then \( \text{Area} = 6 \times 5 \times 15 \times 0.5 = 225 \, \text{cm}^2 \).

#### Circle
- Formula: \( \text{Area} = \pi \times r^2 \)
- Explanation: The area of a circle is calculated by multiplying \( \pi \) (approximately 3.14) by the square of the radius.
- Example: If \( r = 5 \, \text{cm} \), then \( \text{Area} = 3.14 \times 5^2 = 78.5 \, \text{cm}^2 \).

#### Ellipse
- Formula: \( \text{Area} = \pi \times a \times b \)
- Explanation: The area of an ellipse is calculated by multiplying \( \pi \) by the semi-major axis (\( a \)) and the semi-minor axis (\( b \)).
- Example: If \( a = 6 \, \text{cm} \) and \( b = 4 \, \text{cm} \), then \( \text{Area} = 3.14 \times 6 \times 4 = 75.36 \, \text{cm}^2 \).

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2. Three-Dimensional Shapes (Surface Area and Volume)

#### Cube
- Surface Area: \( \text{Surface Area} = 6 \times a^2 \)
- Explanation: A cube has 6 faces, each of which is a square with area \( a^2 \). Multiply by 6 to get the total surface area.
- Example: If \( a = 5 \, \text{cm} \), then \( \text{Surface Area} = 6 \times 5^2 = 150 \, \text{cm}^2 \).

- Volume: \( \text{Volume} = a^3 \) or \( a \times a \times a \)
- Explanation: The volume of a cube is calculated by cubing the length of one of its sides.
- Example: If \( a = 5 \, \text{cm} \), then \( \text{Volume} = 5^3 = 125 \, \text{cm}^3 \).

#### Prism
- Surface Area: \( \text{Surface Area} = 2 \times ba + la \)
- Explanation: The surface area of a prism includes the areas of the two bases (\( 2 \times ba \)) and the lateral area (\( la \)).
- Example: If \( ba = 20 \, \text{cm}^2 \) and \( la = 60 \, \text{cm}^2 \), then \( \text{Surface Area} = 2 \times 20 + 60 = 100 \, \text{cm}^2 \).

- Volume: \( \text{Volume} = ba \times h \)
- Explanation: The volume of a prism is calculated by multiplying the base area (\( ba \)) by the height (\( h \)).
- Example: If \( ba = 20 \, \text{cm}^2 \) and \( h = 5 \, \text{cm} \), then \( \text{Volume} = 20 \times 5 = 100 \, \text{cm}^3 \).

#### Pyramid
- Surface Area: \( \text{Surface Area} = ba + la \)
- Explanation: The surface area of a pyramid includes the base area (\( ba \)) and the lateral area (\( la \)).
- Example: If \( ba = 16 \, \text{cm}^2 \) and \( la = 60 \, \text{cm}^2 \), then \( \text{Surface Area} = 16 + 60 = 76 \, \text{cm}^2 \).

- Volume: \( \text{Volume} = ba \times h \times \frac{1}{3} \)
- Explanation: The volume of a pyramid is one-third of the product of the base area (\( ba \)) and the height (\( h \)).
- Example: If \( ba = 16 \, \text{cm}^2 \) and \( h = 9 \, \text{cm} \), then \( \text{Volume} = 16 \times 9 \times \frac{1}{3} = 48 \, \text{cm}^3 \).

#### Regular Polyhedron
- Surface Area: \( \text{Surface Area} = fa \times s \)
- Explanation: The surface area of a regular polyhedron is calculated by multiplying the area of one face (\( fa \)) by the number of faces (\( s \)).
- Example: If \( fa = 200 \, \text{cm}^2 \) and \( s = 12 \), then \( \text{Surface Area} = 200 \times 12 = 2400 \, \text{cm}^2 \).

- Volume: No simple generic formula is provided for regular polyhedra.

#### Sphere
- Surface Area: \( \text{Surface Area} = 4 \times \pi \times r^2 \)
- Explanation: The surface area of a sphere is calculated by multiplying 4, \( \pi \), and the square of the radius.
- Example: If \( r = 4.5 \, \text{cm} \), then \( \text{Surface Area} = 4 \times 3.14 \times 4.5^2 = 254.5 \, \text{cm}^2 \) (approx.).

- Volume: \( \text{Volume} = \frac{4}{3} \times \pi \times r^3 \)
- Explanation: The volume of a sphere is calculated by multiplying \( \frac{4}{3} \), \( \pi \), and the cube of the radius.
- Example: If \( r = 4.5 \, \text{cm} \), then \( \text{Volume} = \frac{4}{3} \times 3.14 \times 4.5^3 = 381.5 \, \text{cm}^3 \) (approx.).

#### Cylinder
- Surface Area: \( \text{Surface Area} = 2 \pi rh + 2 \pi r^2 \)
- Explanation: The surface area of a cylinder includes the areas of the two circular bases (\( 2 \pi r^2 \)) and the lateral surface area (\( 2 \pi rh \)).
- Example: If \( r = 5 \, \text{cm} \) and \( h = 10 \, \text{cm} \), then \( \text{Surface Area} = 2 \times 3.14 \times 5 \times 10 + 2 \times 3.14 \times 5^2 = 471 \, \text{cm}^2 \).

- Volume: \( \text{Volume} = \pi \times r^2 \times h \)
- Explanation: The volume of a cylinder is calculated by multiplying \( \pi \), the square of the radius, and the height.
- Example: If \( r = 5 \, \text{cm} \) and \( h = 10 \, \text{cm} \), then \( \text{Volume} = 3.14 \times 5^2 \times 10 = 785 \, \text{cm}^3 \).

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Summary

The reference sheet provides formulas and examples for calculating the Area, Surface Area, and Volume of various shapes. Each section clearly explains the formula and demonstrates its application with numerical examples.

Final Answer

\[
\boxed{\text{Refer to the detailed explanations above for each shape's formulas and examples.}}
\]