Geometry and Measurement

Master practical geometry for real-world applications: calculating area, perimeter, volume, and measurements for home projects, construction, landscaping, and spatial planning.

Perimeter (Distance Around)

Perimeter: The total distance around a shape's outside edge.

Rectangle and Square

Rectangle: P = 2l + 2w or P = 2(l + w)

• l = length, w = width

Square: P = 4s

• s = side length

Example: Room is 12 ft × 15 ft, need baseboard

• Perimeter: 2(12) + 2(15) = 24 + 30 = 54 feet
• Minus 3 ft for door: 51 feet of baseboard

Example: Fence around square yard, 40 ft sides

• Perimeter: 4 × 40 = 160 feet

Triangle

Formula: P = a + b + c

• Sum of all three sides

Example: Garden bed with sides 8 ft, 10 ft, 12 ft

• Perimeter: 8 + 10 + 12 = 30 feet
• For edging material

Circle (Circumference)

Formula: C = 2πr or C = πd

• r = radius, d = diameter
• π ≈ 3.14159

Example: Circular patio, 12 ft diameter

• Circumference: π × 12 = 3.14 × 12 ≈ 37.7 feet
• For border stones

Example: Wheel with 26-inch diameter, one complete rotation

• Distance: π × 26 ≈ 81.7 inches ≈ 6.8 feet

Area (Surface Coverage)

Area: The amount of space inside a 2D shape.

Rectangle and Square

Rectangle: A = l × w
Square: A = s²

Example: Room 14 ft × 18 ft, need carpet

• Area: 14 × 18 = 252 square feet

Example: Tile is 12 inch × 12 inch (1 ft × 1 ft)

• Tiles needed: 252 tiles

Triangle

Formula: A = ½ × base × height

Example: Triangular garden, base 10 ft, height 6 ft

• Area: ½ × 10 × 6 = 30 square feet

Note: Height must be perpendicular to base

Circle

Formula: A = πr²

Example: Circular pool, radius 8 feet

• Area: π × 8² = 3.14 × 64 ≈ 201 square feet

Example: Pizza, 16-inch diameter (8-inch radius)

• Area: π × 8² ≈ 201 square inches

Trapezoid

Formula: A = ½ × (base₁ + base₂) × height

Example: Trapezoid lot, bases 50 ft and 70 ft, height 40 ft

• Area: ½ × (50 + 70) × 40 = ½ × 120 × 40 = 2,400 sq ft

Composite Shapes

Break complex shapes into simpler parts.

Example: L-shaped room

• Part 1: 10 ft × 12 ft = 120 sq ft
• Part 2: 8 ft × 6 ft = 48 sq ft
• Total: 120 + 48 = 168 sq ft

Surface Area (3D Shapes)

Surface Area: Total area of all surfaces of a 3D object.

Rectangular Prism (Box)

Formula: SA = 2(lw + lh + wh)

Example: Box 5 ft × 3 ft × 4 ft, need paint

• SA = 2(5×3 + 5×4 + 3×4)
• SA = 2(15 + 20 + 12) = 2(47) = 94 sq ft

Cube

Formula: SA = 6s²

Example: Cube storage unit, 3 ft sides

• SA = 6 × 3² = 6 × 9 = 54 sq ft

Cylinder

Formula: SA = 2πr² + 2πrh

• Two circles + rectangle wrapped around

Example: Water tank, radius 4 ft, height 10 ft

• Circles: 2 × π × 4² = 2 × 3.14 × 16 ≈ 100.5 sq ft
• Side: 2 × π × 4 × 10 ≈ 251.3 sq ft
• Total: ≈ 352 sq ft

Volume (3D Space)

Volume: The amount of 3D space inside an object.

Rectangular Prism

Formula: V = l × w × h

Example: Moving box 2 ft × 3 ft × 4 ft

• Volume: 2 × 3 × 4 = 24 cubic feet

Example: Room 12 ft × 15 ft × 8 ft high

• Volume: 12 × 15 × 8 = 1,440 cubic feet

Cube

Formula: V = s³

Example: Cubic container, 5 ft sides

• Volume: 5³ = 125 cubic feet

Cylinder

Formula: V = πr²h

Example: Hot tub, radius 4 ft, depth 3 ft

• Volume: π × 4² × 3 = 3.14 × 16 × 3 ≈ 150.7 cubic feet

Convert to gallons: 1 cubic foot ≈ 7.48 gallons

• Gallons: 150.7 × 7.48 ≈ 1,127 gallons

Sphere

Formula: V = (4/3)πr³

Example: Basketball, radius 4.7 inches

• Volume: (4/3) × π × 4.7³ ≈ 435 cubic inches

Cone

Formula: V = (1/3)πr²h

Example: Conical pile of sand, radius 6 ft, height 8 ft

• Volume: (1/3) × π × 6² × 8 ≈ 301.6 cubic feet

Pyramid

Formula: V = (1/3) × Base Area × height

Example: Square pyramid, base 10 ft × 10 ft, height 12 ft

• Base area: 10 × 10 = 100 sq ft
• Volume: (1/3) × 100 × 12 = 400 cubic feet

Practical Applications

Painting

Walls: Calculate area, subtract doors/windows

Example: Wall 20 ft wide × 8 ft high

• Area: 20 × 8 = 160 sq ft
• Door: 3 ft × 7 ft = 21 sq ft
• Paintable: 160 − 21 = 139 sq ft

Coverage: Most paint covers 350-400 sq ft/gallon

Total walls: 139 + 180 + 139 + 180 = 638 sq ft

• Gallons needed: 638 ÷ 350 ≈ 1.8 → Buy 2 gallons

Flooring

Example: 15 ft × 18 ft room

• Area: 270 sq ft
• Add 10% waste: 270 × 1.10 = 297 sq ft
• Order 300 sq ft

Cost: Flooring at $4.50/sq ft

• Total: 300 × $4.50 = $1,350

Concrete/Mulch

Calculate volume, then convert to needed measurement.

Concrete Patio: 12 ft × 16 ft × 4 inches deep

• Convert depth: 4 inches = 0.33 ft
• Volume: 12 × 16 × 0.33 = 63.4 cubic feet
• Convert to yards: 63.4 ÷ 27 = 2.35 cubic yards
• Order 2.5 cubic yards

Mulch for Garden Bed: 10 ft × 4 ft × 3 inches deep

• Convert depth: 3 inches = 0.25 ft
• Volume: 10 × 4 × 0.25 = 10 cubic feet
• Mulch sold in 2 cubic foot bags
• Need 5 bags

Fencing

Calculate perimeter, account for gates.

Example: Rectangular yard 50 ft × 80 ft

• Perimeter: 2(50 + 80) = 260 feet
• Gate: 4 ft
• Fencing: 260 − 4 = 256 feet
• Posts every 8 ft: 256 ÷ 8 = 32 posts

Wallpaper

Formula: Rolls = (Wall Height × Perimeter) / Roll Coverage

Example: Room perimeter 48 ft, height 8 ft

• Area: 48 × 8 = 384 sq ft
• Roll covers 30 sq ft
• Rolls: 384 ÷ 30 = 12.8 → Buy 13 rolls

Swimming Pool

Rectangular pool: 20 ft × 40 ft × 6 ft deep

• Volume: 20 × 40 × 6 = 4,800 cubic feet
• Gallons: 4,800 × 7.48 = 35,904 gallons

Circular pool: 24 ft diameter (12 ft radius), 4 ft deep

• Volume: π × 12² × 4 ≈ 1,810 cubic feet
• Gallons: 1,810 × 7.48 ≈ 13,539 gallons

Unit Conversions for Area and Volume

Area Conversions

Example: 2,160 sq inches to sq feet

• 2,160 ÷ 144 = 15 sq ft

Volume Conversions

Example: 5 cubic yards to cubic feet

• 5 × 27 = 135 cubic feet

Pythagorean Theorem

For right triangles: a² + b² = c²

• c = hypotenuse (longest side)
• a, b = other two sides

Finding Diagonal

Example: TV mounting on wall, 60 inches wide × 36 inches high

• Diagonal: √(60² + 36²) = √(3,600 + 1,296) = √4,896 ≈ 70 inches

Example: Rectangular room 12 ft × 16 ft, diagonal distance

• Diagonal: √(12² + 16²) = √(144 + 256) = √400 = 20 feet

Checking Square

Carpentry rule: 3-4-5 triangle (or multiples)

Example: Deck corner, measure 6 ft one side, 8 ft other side

• Diagonal should be: √(6² + 8²) = √(36 + 64) = √100 = 10 feet
• If diagonal = 10 ft, corner is square (90°)

Scale and Maps

Scale ratio: 1:100 means 1 unit on drawing = 100 units in reality

Example: Blueprint scale 1:50

• Drawing shows 3 inches
• Reality: 3 × 50 = 150 inches = 12.5 feet

Map scale: 1 inch = 10 miles

• Cities are 4.5 inches apart on map
• Actual distance: 4.5 × 10 = 45 miles

Practice Problems

Perimeter

1. Rectangular yard 35 ft × 50 ft. How much fencing needed (with 4 ft gate)?
2. Circular flower bed, radius 5 ft. Perimeter for edging?

Area

3. Room 16 ft × 20 ft. How many sq ft of carpet?
4. Circular patio, diameter 18 ft. Area?

Volume

5. Rectangular aquarium 4 ft × 2 ft × 2.5 ft. Volume in cubic feet and gallons?
6. Cylindrical water tank, radius 3 ft, height 8 ft. Volume?

Applications

7. Wall 25 ft × 9 ft, door 3 ft × 7 ft, window 4 ft × 5 ft. Paintable area?
8. Concrete slab 15 ft × 20 ft × 4 inches. Volume in cubic yards?
9. Room 14 ft × 18 ft, need diagonal measurement.

Conversions

10. Convert 2,000 sq ft to sq yards
11. Convert 100 cubic feet to gallons

Solutions

1. 166 ft (2(35+50) − 4)
2. 31.4 ft (2 × π × 5)
3. 320 sq ft (16 × 20)
4. 254.5 sq ft (π × 9²)
5. 20 cu ft (4 × 2 × 2.5), 149.6 gallons
6. 226.2 cu ft (π × 3² × 8)
7. 179 sq ft (225 − 21 − 20)
8. 3.7 cu yd (15 × 20 × 0.33 ÷ 27)
9. 22.8 ft (√(14² + 18²))
10. 222.2 sq yd (2000 ÷ 9)
11. 748 gallons (100 × 7.48)

Key Takeaways

✓ Know your formulas: memorize area and volume for common shapes
✓ Add waste: always add 10-15% for cuts and mistakes
✓ Convert units carefully: keep track of feet vs inches, sq ft vs sq yards
✓ Pythagorean theorem: essential for diagonals and checking square
✓ Visualize: break complex shapes into simple parts
✓ Measure twice: errors in measurement are costly

Real-World Applications

• Home Improvement: Calculate materials for painting, flooring, fencing
• Landscaping: Determine mulch, stone, or sod needed
• Construction: Estimate concrete, lumber, roofing
• Interior Design: Plan furniture arrangement, window treatments
• Moving: Choose appropriate truck/storage size
• DIY Projects: Calculate materials accurately

Next Steps

Move to Chapter 09: Estimation & Mental Math to learn quick calculation techniques, approximation methods, and mental math tricks for everyday situations.