Area and Perimeter formulas for common geometric shapes.

The image you provided is a summary of formulas for calculating the area and perimeter/circumference of various geometric shapes. Below, I will explain each formula and provide examples to illustrate how they are used.

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1. Square

- Area: \( A = s^2 \)
- The area of a square is calculated by squaring the length of one of its sides.
- Perimeter: \( P = 4s \)
- The perimeter of a square is calculated by multiplying the length of one side by 4.

Example:
If the side length \( s = 5 \) units:
- Area: \( A = s^2 = 5^2 = 25 \) square units
- Perimeter: \( P = 4s = 4 \times 5 = 20 \) units

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2. Rectangle

- Area: \( A = l \times w \)
- The area of a rectangle is calculated by multiplying its length (\( l \)) by its width (\( w \)).
- Perimeter: \( P = 2l + 2w \)
- The perimeter of a rectangle is calculated by adding twice the length and twice the width.

Example:
If the length \( l = 6 \) units and the width \( w = 4 \) units:
- Area: \( A = l \times w = 6 \times 4 = 24 \) square units
- Perimeter: \( P = 2l + 2w = 2 \times 6 + 2 \times 4 = 12 + 8 = 20 \) units

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3. Triangle

- Area: \( A = \frac{1}{2} \times b \times h \)
- The area of a triangle is calculated by multiplying half of its base (\( b \)) by its height (\( h \)).

Example:
If the base \( b = 8 \) units and the height \( h = 5 \) units:
- Area: \( A = \frac{1}{2} \times b \times h = \frac{1}{2} \times 8 \times 5 = 20 \) square units

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4. Parallelogram

- Area: \( A = b \times h \)
- The area of a parallelogram is calculated by multiplying its base (\( b \)) by its height (\( h \)).

Example:
If the base \( b = 7 \) units and the height \( h = 3 \) units:
- Area: \( A = b \times h = 7 \times 3 = 21 \) square units

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5. Trapezoid

- Area: \( A = \frac{1}{2} \times (b_1 + b_2) \times h \)
- The area of a trapezoid is calculated by taking half the sum of its two parallel bases (\( b_1 \) and \( b_2 \)) and multiplying it by the height (\( h \)).

Example:
If the bases are \( b_1 = 6 \) units, \( b_2 = 10 \) units, and the height \( h = 4 \) units:
- Area: \( A = \frac{1}{2} \times (b_1 + b_2) \times h = \frac{1}{2} \times (6 + 10) \times 4 = \frac{1}{2} \times 16 \times 4 = 32 \) square units

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6. Circle

- Area: \( A = \pi r^2 \)
- The area of a circle is calculated by multiplying pi (\( \pi \)) by the radius squared (\( r^2 \)).
- Circumference: \( C = 2\pi r \) or \( C = \pi d \)
- The circumference of a circle is calculated by multiplying \( 2\pi \) by the radius (\( r \)), or by multiplying \( \pi \) by the diameter (\( d \)).

Example:
If the radius \( r = 3 \) units:
- Area: \( A = \pi r^2 = \pi \times 3^2 = 9\pi \) square units (or approximately \( 28.27 \) square units)
- Circumference: \( C = 2\pi r = 2\pi \times 3 = 6\pi \) units (or approximately \( 18.85 \) units)

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Summary of Formulas

1. Square:
- Area: \( A = s^2 \)
- Perimeter: \( P = 4s \)

2. Rectangle:
- Area: \( A = l \times w \)
- Perimeter: \( P = 2l + 2w \)

3. Triangle:
- Area: \( A = \frac{1}{2} \times b \times h \)

4. Parallelogram:
- Area: \( A = b \times h \)

5. Trapezoid:
- Area: \( A = \frac{1}{2} \times (b_1 + b_2) \times h \)

6. Circle:
- Area: \( A = \pi r^2 \)
- Circumference: \( C = 2\pi r \) or \( C = \pi d \)

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Final Answer

The formulas provided in the image are essential tools for calculating the area and perimeter/circumference of various shapes. Each formula is derived from the geometric properties of the shape and can be applied as demonstrated in the examples above.

Boxed Final Answer:
\[
\boxed{\text{See explanations and examples above.}}
\]