Area and Volume

• Area is the measure of the two-dimensional space inside a closed shape

Area of Basic Shapes

• Rectangle: Area = length × width
• Square: Area = side × side = side²
• Triangle: Area = 1/2 × base × height
• Circle: Area = π × radius² (π ≈ 3.14159)
• Parallelogram: Area = base × height
• Trapezoid: Area = 1/2 × (base1 + base2) × height

Units of Area

• Area is measured in square units (e.g., square meters, square feet, square inches)
• Unit conversions are important (e.g., 1 square foot = 144 square inches)

Volume

• Volume is the measure of the three-dimensional space occupied by an object

Volume of Basic Shapes

• Cube: Volume = side³
• Rectangular Prism: Volume = length × width × height
• Sphere: Volume = (4/3) × π × radius³
• Cylinder: Volume = π × radius² × height
• Cone: Volume = (1/3) × π × radius² × height
• Pyramid: Volume = (1/3) × base area × height

Units of Volume

• Volume is measured in cubic units (e.g., cubic meters, cubic feet, cubic centimeters)
• Common conversions include: 1 liter = 1000 cubic centimeters

Surface Area

• Surface area is the total area of all the surfaces of a 3D object

Surface Area of Basic Shapes

• Cube: Surface Area = 6 × side²
• Rectangular Prism: Surface Area = 2 × (length × width + length × height + width × height)
• Sphere: Surface Area = 4 × π × radius²
• Cylinder: Surface Area = 2 × π × radius² + 2 × π × radius × height (including both ends)
• Cone: Surface Area = π × radius × (radius + slant height)

Calculating Area of Irregular Shapes

• Divide the irregular shape into smaller, regular shapes
• Calculate the area of each regular shape
• Sum the areas to find the total area

Calculating Volume of Irregular Objects

• Use displacement method (Archimedes' principle)
• Submerge the object in water and measure the volume of water displaced
• Water displacement equals the volume of the object
• For some complex shapes, integral calculus can be used

Area and Volume Relationships

• Understanding how changes in dimensions affect area and volume
• Doubling the side of a square quadruples its area
• Doubling the side of a cube increases its volume by a factor of eight

Practical Applications of Area and Volume

• Calculating paint needed for a wall (area)
• Determining the amount of liquid a container can hold (volume)
• Estimating materials for construction projects

Key Formulas for Area

• Square: A = s² (s = side length)
• Rectangle: A = lw (l = length, w = width)
• Triangle: A = 0.5bh (b = base, h = height)
• Circle: A = πr² (r = radius)
• Parallelogram: A = bh (b = base, h = height)
• Trapezoid: A = 0.5(b1 + b2)h (b1, b2 = base lengths, h = height)

Example Problems: Area

• Problem: Find the area of a rectangle with length 10 cm and width 5 cm
• Solution: A = lw = 10 cm * 5 cm = 50 cm²
• Problem: Calculate the area of a circle with radius 7 meters
• Solution: A = πr² = π * (7 m)² ≈ 153.94 m²
• Problem: Find the height of a triangle with area 24 cm² and base 8 cm
• Solution: A = 0.5bh => 24 cm² = 0.5 * 8 cm * h => h = 6 cm

Key Formulas for Volume

• Cube: V = s³ (s = side length)
• Rectangular Prism: V = lwh (l = length, w = width, h = height)
• Sphere: V = (4/3)πr³ (r = radius)
• Cylinder: V = πr²h (r = radius, h = height)
• Cone: V = (1/3)πr²h (r = radius, h = height)
• Pyramid: V = (1/3)Bh (B = base area, h = height)

Example Problems: Volume

• Problem: Find the volume of a cube with side length 4 inches
• Solution: V = s³ = (4 in)³ = 64 in³
• Problem: Calculate the volume of a cylinder with radius 3 cm and height 10 cm
• Solution: V = πr²h = π * (3 cm)² * 10 cm ≈ 282.74 cm³
• Problem: Compute the volume of a sphere with radius 6 meters
• Solution: V = (4/3)πr³ = (4/3)π * (6 m)³ ≈ 904.78 m³

Key Formulas for Surface Area

• Cube: SA = 6s² (s = side length)
• Rectangular Prism: SA = 2(lw + lh + wh) (l = length, w = width, h = height)
• Sphere: SA = 4πr² (r = radius)
• Cylinder: SA = 2πr² + 2πrh (r = radius, h = height)
• Cone: SA = πr(r + √(h² + r²)) (r = radius, h = height)

Example Problems: Surface Area

• Problem: Calculate the surface area of a cube with side length 5 cm.
• Solution: SA = 6s² = 6 * (5 cm)² = 150 cm²
• Problem: Find the surface area of a cylinder with radius 2 m and height 8 m.
• Solution: SA = 2πr² + 2πrh = 2π * (2 m)² + 2π * (2 m) * (8 m) ≈ 125.66 m²
• Problem: Determine the surface area of a sphere with radius 3 m.
• Solution: SA = 4πr² = 4π * (3 m)² ≈ 113.10 m²

Practical Volume Calculations

• Volume of a room (rectangular prism): V = length × width × height
• Volume of liquid in a cylindrical tank: V = π × radius² × height
• Estimating cubic yards of concrete needed for a slab: Volume = length × width × thickness (convert to yards)

Perimeter vs Area

• Perimeter is the length of the boundary of a two-dimensional shape.
• Area is the amount of space enclosed within that boundary.
• Perimeter is measured in linear units (e.g., meters, feet).
• Area is measured in square units (e.g., square meters, square feet).

Scale Factors and Area/Volume

• If a shape is scaled by a factor of 'k', its area is scaled by k².
• If a shape is scaled by a factor of 'k', its volume is scaled by k³.
• Example: If a square's side length is doubled (k=2), its area becomes 4 times larger (2²).

Area of a sector

• Area of a sector = (θ/360) * πr²
• θ = angle of the sector in degrees
• r = radius of the circle

Area of an Annulus (Ring)

• Area of an annulus = π(R² - r²)
• R = radius of the outer circle
• r = radius of the inner circle

Cavalieri's Principle

• If two solids have the same height and the same cross-sectional area at every level, then they have the same volume.
• Useful for finding the volume of oblique shapes.

Area and Volume in Calculus

• Area between curves: Integrate the difference of the functions over the interval [a, b].
• Volume of revolution (disk method, shell method): Integrate π[f(x)]² or 2πx f(x) over an interval.

Estimating area using grids

• Overlay a grid of squares on the shape
• Count the squares that are fully inside the shape
• Estimate the area of partially filled squares
• Sum the areas to find the total estimated area