Area and Volume Calculations: Rectangles, Triangles, and Cubes

Area: General Concept

Area is the measurement of a two-dimensional space occupied by a certain shape. It is the amount of surface enclosed within the boundaries of this shape. Area is measured in square units such as (cm²❓, m², ft², yd², etc.).

Rectangle Area

1. Rectangle Definition:
   A rectangle is a quadrilateral in which each pair of opposite sides is equal and parallel, and all four angles are right angles (90 degrees).

2. Area Calculation:
   To calculate the area of a rectangle, we multiply its length by its width:

   <a data-bs-toggle="modal" data-bs-target="#questionModal-108879" role="button" aria-label="Open Question" class="keyword-wrapper question-trigger"><span class="keyword-container">A = <a data-bs-toggle="modal" data-bs-target="#questionModal-382385" role="button" aria-label="Open Question" class="keyword-wrapper question-trigger"><span class="keyword-container">l x w</span><span class="flag-trigger">❓</span></a></span><span class="flag-trigger">❓</span></a>

   Where:

   • A: Area
   • L: Length
   • W: Width

3. Units:
   The length and width must be in the same unit before calculating the area.

4. Practical Examples:

   • Example 1:
     A rectangular piece of land is 20 meters long and 10 meters wide. Its area is:
     A = L x W = 20 m x 10 m = 200 m²

   • Example 2:
     A rectangular room is 5 meters by 4 meters. To cover the floor with square tiles of 20 cm side length, the number of tiles needed is:
     Area of the floor: A = 5 m x 4 m = 20 m²
     Area of one tile: A = 20 cm x 20 cm = 400 <a data-bs-toggle="modal" data-bs-target="#questionModal-382383" role="button" aria-label="Open Question" class="keyword-wrapper question-trigger"><span class="keyword-container"><a data-bs-toggle="modal" data-bs-target="#questionModal-382382" role="button" aria-label="Open Question" class="keyword-wrapper question-trigger"><span class="keyword-container">cm²</span><span class="flag-trigger">❓</span></a></span><span class="flag-trigger">❓</span></a>
     Convert area of the floor to cm²: 20 m² = 20 x 10,000 cm² = 200,000 cm²
     Number of tiles: Number of tiles = 200,000 cm² / 400 cm² = 500 tiles

Triangle Area

1. Triangle Definition:
   A triangle is a geometric shape consisting of three sides and three angles.

2. Area Calculation:
   To calculate the area of a triangle, we multiply one-half of the base length by the height:

   A = 1/2 x B x H

   Where:

   • A: Area
   • B: Base - the length of one side of the triangle.
   • H: Height - the perpendicular distance from the base to the opposite vertex.

3. Important Notes:

   • In a right-angled triangle, one of the sides of the right angle can be considered the base and the other the height.
   • The height must be perpendicular to the base (forming a 90-degree angle).

4. Practical Examples:

   • Example 1:
     A triangular piece of land has a base of 15 meters and a height of 8 meters. Its area is:
     A = 1/2 x B x H = 1/2 x 15 m x 8 m = 60 m²

   • Example 2:
     A right-angled triangle has sides of 6 cm❓❓ and 8 cm forming the right angle. Its area is:
     A = 1/2 x B x H = 1/2 x 6 cm x 8 cm = 24 cm²

Area Units and Conversion

1. Common Units:

   • Square centimeter (cm²)
   • Square meter (m²)
   • Square foot (ft²)
   • Square inch (in²)
   • Square yard (yd²)
   • Square mile (mi²)
   • Acre - approximately equals 43,560 square feet

2. Unit Conversions:

   • 1 square meter = 10,000 square centimeters
   • 1 square foot = 144 square inches
   • 1 square yard = 9 square feet
   • 1 acre = 43,560 square feet

3. Conversion Examples:

   • Example 1: Convert 5 square meters to square centimeters.
     5 m² = 5 x 10,000 cm² = 50,000 cm²
   • Example 2: Convert 1000 square feet to square yards.
     1000 ft² = 1000 / 9 yd² = 111.11 yd² (approximately)

Volume: General Concept

Volume is the measurement of a three-dimensional space occupied by a certain object. In other words, it is the amount of space enclosed within the boundaries of this object. Volume is measured in cubic units such as (cm³❓, m³, ft³, etc.).

Cube Volume

1. Cube Definition:
   A cube is a three-dimensional shape with six identical square faces, and all its angles are right angles.

2. Volume Calculation:
   To calculate the volume of a cube, we multiply the length of its side by itself three times (cube the side length):

   V = L x L x L = L³

   Where:

   • V: Volume
   • L: Length of side

3. Units:
   The side length must be in an appropriate unit of measurement (cm, m, ft, etc.).

4. Practical Examples:

   • Example 1:
     A cube has a side length of 5 cm. Its volume is:
     V = L³ = 5 cm x 5 cm x 5 cm = 125 cm³

   • Example 2:
     A box in the shape of a cube has a volume of 64 cubic meters. The length of the side of the box is:
     Since V = L³, then L = ³√V = ³√64 = 4 m

Volume Units and Conversion

1. Common Units:

   • Cubic centimeter (cm³)
   • Cubic meter (m³)
   • Cubic foot (ft³)
   • Cubic inch (in³)
   • Liter (L) - approximately equals 1000 cubic centimeters

2. Unit Conversions:

   • 1 cubic meter = 1,000,000 cubic centimeters
   • 1 cubic foot = 1728 cubic inches
   • 1 liter = 1000 cubic centimeters

3. Conversion Examples:

   • Example 1: Convert 2 cubic meters to cubic centimeters.
     2 m³ = 2 x 1,000,000 cm³ = 2,000,000 cm³
   • Example 2: Convert 5000 cubic centimeters to liters.
     5000 cm³ = 5000 / 1000 L = 5 L