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02-04 - Perimeter, Area, Volume

Phase: 2 | Subject: 02-04 Prerequisites: 02-03-polygons-and-circles.md Next subject: 02-05-pythagoras-and-right-triangle-trig.md

Learning Objectives

By the end of this subject, you will be able to:

  1. Calculate perimeter of composite shapes
  2. Calculate area of triangles, quadrilaterals, and circles
  3. Calculate surface area of prisms, pyramids, cylinders, cones, and spheres
  4. Calculate volume of the same solids
  5. Decompose composite solids into simpler shapes

Core Content

Perimeter

Perimeter is the total distance around a shape.

Common shapes: - Rectangle: P = 2(l + w) - Square: P = 4s - Circle: P = 2πr (circumference) - Triangle: P = a + b + c

Composite shapes: Add the outer edges only (don't count internal lines twice).

Area Formulas

Shape Area Formula
Square A = s²
Rectangle A = lw
Triangle A = (1/2)bh
Parallelogram A = bh
Trapezium A = (1/2)(a + b)h
Circle A = πr²

Composite Areas

Break the shape into simpler parts, calculate each, then add (or subtract).

Example: L-shaped room

┌──────┐
│      │ 5m
│      ├────┐
│      │    │ 3m
3m     │    │
│      │    │
└──────┴────┘
   8m

Split into two rectangles: - Left: 3m × 5m = 15 m² - Right: 5m × 3m = 15 m² - Total: 30 m²

Surface Area

Surface area is the total area of all faces of a 3D object.

Prism

A prism has two identical parallel ends and rectangular sides.

$Surface Area = 2 × (area of base) + (perimeter of base) × height
SA = 2B + Ph
$

Pyramid

$Surface Area = (area of base) + (area of triangular faces)
$

For a regular pyramid with n triangular faces: SA = B + (1/2) × P × l where l = slant height

Cylinder

$SA = 2πr² + 2πrh = 2πr(r + h)
$

Two circular ends + curved side (unrolled = rectangle 2πr × h)

Cone

$SA = πr² + πrl = πr(r + l)
$

where l = slant height

Sphere

$SA = 4πr²
$

Volume

Volume measures how much space a 3D object occupies.

Prism

$Volume = (area of base) × height
V = Bh
$

Pyramid

$Volume = (1/3) × (area of base) × height
V = (1/3)Bh
$

Cylinder

$Volume = πr²h
$

Cone

$Volume = (1/3)πr²h
$

Sphere

$Volume = (4/3)πr³
$

⚠️ THIS IS CRITICAL — V = (4/3)πr³ for spheres appears throughout physics (planets, drops, bubbles) and is one of the few non-integer coefficients in geometry. The 4/3 comes from calculus integration.

Key Relationship

Cone volume = (1/3) × cylinder volume with same base and height. Pyramid volume = (1/3) × prism volume with same base and height.

Key Terms

Worked Examples

Example 1: Composite area

Shape consists of a rectangle (6m × 4m) with a semicircle (diameter 4m) attached to one side.

  1. Rectangle area: 6 × 4 = 24 m²
  2. Semicircle area: (1/2)π(2²) = 2π ≈ 6.28 m²
  3. Total: 24 + 2π ≈ 30.28 m²

Example 2: Surface area of cylinder

Cylinder with radius 3cm and height 10cm.

  1. Two circles: 2 × π(3²) = 18π cm²
  2. Curved surface: 2π(3)(10) = 60π cm²
  3. Total: 78π ≈ 245.04 cm²

Example 3: Volume of cone

Cone with radius 5cm and height 12cm.

  1. V = (1/3)π(5²)(12)
  2. V = (1/3)π(25)(12)
  3. V = 100π ≈ 314.16 cm³

Quiz

Q1: What does the concept of Area Formulas primarily refer to in this subject?

A) A visual representation of Area Formulas B) The definition and application of Area Formulas C) A computational error related to Area Formulas D) A historical anecdote about Area Formulas

Correct: B)

Q2: What is the primary purpose of Circle?

A) It is used only in advanced research contexts B) It is primarily a historical notation system C) It is used to circle in mathematical analysis D) It replaces all other methods in this domain

Correct: C)

Q3: Which statement about Composite Areas is TRUE?

A) Composite Areas is an advanced topic beyond this subject's scope B) Composite Areas is mentioned only as a historical footnote C) Composite Areas is not related to this subject D) Composite Areas is a fundamental concept covered in this subject

Correct: D)

Q4: Based on the worked examples in this subject, what is the correct result?

A) 26 m B) A different result from a common mistake C) The inverse of the correct answer D) An unrelated numerical value

Correct: A)

Q5: How are Composite Areas and Cylinder related?

A) Composite Areas and Cylinder are closely related concepts B) Composite Areas and Cylinder are completely unrelated topics C) Composite Areas is a special case of Cylinder D) Composite Areas is the inverse of Cylinder

Correct: A)

Q6: What is a common pitfall when working with Perimeter?

A) Perimeter is always computed the same way in all contexts B) The main error with Perimeter is using it when it is not needed C) A common mistake is confusing Perimeter with a similar concept D) Perimeter has no common misconceptions

Correct: C)

Q7: When should you apply Surface Area?

A) Surface Area is not practically useful B) Use Surface Area only in pure mathematics contexts C) Avoid Surface Area unless explicitly instructed D) Apply Surface Area to solve problems in this subject's domain

Correct: D)

Practice Problems

  1. Perimeter of rectangle 8m by 5m Answer: 2(8 + 5) = 26 m

  2. Area of trapezium with parallel sides 6m and 10m, height 4m Answer: (1/2)(6 + 10)(4) = 32 m²

  3. Surface area of cube with side 3cm Answer: 6 × (3²) = 54 cm²

  4. Volume of rectangular prism 4m × 3m × 2m Answer: 4 × 3 × 2 = 24 m³

  5. Surface area of sphere with radius 7cm Answer: 4π(49) = 196π ≈ 615.75 cm²

  6. Volume of sphere with radius 6cm Answer: (4/3)π(216) = 288π ≈ 904.78 cm³

  7. Composite shape: rectangle 10m × 6m plus triangle (base 6m, height 4m) on top Answer: Rectangle: 60 m². Triangle: (1/2)(6)(4) = 12 m². Total: 72 m²

Summary

Key takeaways:

Pitfalls

Next Steps

Next up: 02-05-pythagoras-and-right-triangle-trig.md