📐 COMPLETE TEACHING NOTES: PERIMETER, AREA AND VOLUME 📐
For Grade 7 Students - Cambridge AS & A Level Mathematics
🔷 SECTION 7.1: PERIMETER AND AREA IN TWO DIMENSIONS
What is Perimeter?
Perimeter is the total distance around the outside of a flat shape. Think of it like walking around the edge of a park - the distance you walk is the perimeter!
- The length of fence needed around a garden
- The distance around a running track
- The coastline of an island
How to find perimeter: Simply add up all the side lengths of your shape.
What is Area?
Area measures how much space is inside a flat shape. Imagine you're painting a wall - the area tells you how much paint you need!
What does cm² mean? It means "how many 1 cm × 1 cm squares fit inside the shape".
Area Formulas for Common Shapes
1. Rectangle
Area = 7 × 5 = 35 cm²
2. Square
Area = 6 × 6 = 36 cm²
3. Triangle
Area = ½ × 5 × 6 = ½ × 30 = 15 cm²
4. Parallelogram
5. Trapezium
Area = ½ × (4 + 8) × 5 = ½ × 12 × 5 = 30 cm²
Working with Complex Shapes
Sometimes you'll see shapes that look complicated. The secret is to break them into simpler shapes!
- Divide the complex shape into rectangles, triangles, or other simple shapes
- Find the area of each simple shape
- Add all the areas together
CIRCLES - A Special Shape
Parts of a Circle
- Radius (r): Distance from center to edge
- Diameter (d): Distance across through center
- d = 2r (diameter is twice the radius)
- Circumference: The perimeter of a circle
The Magic Number π (Pi)
π is a special number approximately equal to 3.141592654...
The decimal places go on forever and never repeat - we call this an irrational number.
For calculations, we usually use the π button on calculators, or approximate it as 3.142.
Circle Formulas
C = π × d (using diameter)
C = 2 × π × r (using radius)
A = π × r² (pi times radius squared)
Circumference:
C = π × d = π × 8 = 25.13... ≈ 25.1 mm (3 significant figures)
Area (need radius first):
radius = diameter ÷ 2 = 8 ÷ 2 = 4 mm
A = π × r² = π × 4² = π × 16 = 50.27... ≈ 50.3 mm² (3 s.f.)
Circumference:
C = 2 × π × r = 2 × π × 5 = 31.42... ≈ 31.4 cm (3 s.f.)
Area:
A = π × r² = π × 5² = π × 25 = 78.54... ≈ 78.5 cm² (3 s.f.)
Exact Answers Using π
Sometimes you're asked for an exact answer. This means you must leave π in your answer.
• Circle with diameter 12 cm: C = 12π cm (exact answer)
• Circle with radius 5 m: A = 25π m² (exact answer)
Working backwards:
• If C = 8π cm, then diameter = 8 cm and radius = 4 cm
• If A = 49π cm², then r² = 49, so radius = 7 cm
Sectors and Arcs
A sector is a "slice" of a circle, like a piece of pizza!
- Arc: The curved edge of the sector
- Angle θ: The angle at the center (measured in degrees)
- Minor sector: The smaller piece
- Major sector: The larger piece
Sector Formulas
Think: "The sector is a fraction of the whole circle"
Area = (θ/360) × π × r²
Arc length = (θ/360) × 2 × π × r
Area:
Area = (30/360) × π × 5²
Area = (1/12) × π × 25
Area = 6.54... ≈ 6.54 m² (3 s.f.)
Perimeter (arc + two straight sides):
Arc length = (30/360) × 2 × π × 5 = 2.62... m
Two radii = 5 + 5 = 10 m
Total perimeter = 2.62 + 10 = 12.6 m (3 s.f.)
🔶 SECTION 7.2: THREE-DIMENSIONAL OBJECTS
Now we move from flat shapes to solid shapes - objects that have length, width, AND height!
What is a Net?
A net is a flat pattern that can be folded to make a 3D shape. Think of how a cardboard box starts flat and then folds up!
When folded: Points marked A join together, Points marked B join together - This creates a cube!
Key 3D Shape Terms
- Face: A flat surface on the solid
- Edge: Where two faces meet (like a line)
- Vertex (plural: vertices): A corner where edges meet
- Surface Area: Total area of all faces
- Volume: Amount of space inside the solid
🔷 SECTION 7.3: SURFACE AREAS AND VOLUMES OF SOLIDS
Understanding Volume Units
If lengths are in centimeters:
- Area is in cm² (square centimeters)
- Volume is in cm³ (cubic centimeters)
What does cm³ mean? How many 1 cm × 1 cm × 1 cm cubes fit inside!
1. CUBOIDS (Rectangular Boxes)
V = a × b × c (length × width × height)
SA = 2(ab + ac + bc)
This formula accounts for all 6 rectangular faces:
- Top and bottom: 2 × (a × b)
- Front and back: 2 × (a × c)
- Left and right sides: 2 × (b × c)
2. PRISMS
A prism is a solid with the same cross-section all along its length.
V = Area of cross-section × length
SA = 2 × (area of cross-section) + (perimeter of cross-section × length)
If triangle has base 10 cm and height 6 cm, and length is 32 cm:
Volume:
Cross-section area = ½ × 10 × 6 = 30 cm²
Volume = 30 × 32 = 960 cm³
3. CYLINDERS
A cylinder is like a prism but with a circular cross-section.
V = π × r² × h
CSA = 2 × π × r × h
TSA = 2πrh + 2πr²
Understanding the Cylinder Net
If you "unwrap" a cylinder, the curved surface becomes a rectangle with:
- Width = circumference of circle = 2πr
- Height = h
- So curved surface area = 2πr × h = 2πrh!
4. PYRAMIDS
A pyramid has a polygon base and triangular sides that meet at a point called the apex.
V = (1/3) × base area × perpendicular height
Surface area: Find the area of the base + area of all triangular faces
Square pyramid with base side 8 cm and perpendicular height 12 cm:
Volume:
Base area = 8 × 8 = 64 cm²
Volume = (1/3) × 64 × 12 = (1/3) × 768 = 256 cm³
5. CONES
A cone is like a pyramid but with a circular base.
V = (1/3) × π × r² × h
CSA = π × r × l
(where l is the slant height)
TSA = πrl + πr²
Volume:
V = (1/3) × π × 4² × 10
V = (1/3) × π × 16 × 10
V = (160/3) × π
V = 167.55... ≈ 168 cm³ (3 s.f.)
6. SPHERES
A sphere is a perfectly round ball.
V = (4/3) × π × r³
SA = 4 × π × r²
Hemisphere (Half a Sphere)
V = (2/3) × π × r³ (half the sphere volume)
CSA = 2 × π × r² (half the sphere surface)
TSA = 2πr² + πr² = 3πr²
Volume:
V = (4/3) × π × 40³
V = (4/3) × π × 64000
V = (256000/3) × π cm³ (exact answer)
Surface area:
SA = 4 × π × 40²
SA = 4 × π × 1600
SA = 6400π cm² (exact answer)
Summary Table of 3D Formulas
| Shape | Volume Formula | Surface Area |
|---|---|---|
| Cuboid | V = l × w × h | SA = 2(lw + lh + wh) |
| Prism | V = (cross-section area) × length | SA = 2A + Pl |
| Cylinder | V = πr²h | SA = 2πrh + 2πr² |
| Pyramid | V = (1/3) × base area × h | Add all face areas |
| Cone | V = (1/3)πr²h | SA = πrl + πr² |
| Sphere | V = (4/3)πr³ | SA = 4πr² |
Problem-Solving Strategies
- Break the shape into simpler parts
- Calculate volume/area of each part
- Add or subtract as needed
- Write down the formula
- Substitute known values
- Rearrange to solve for the unknown
- Check your units (cm, m, mm, etc.)
- For area: use square units (cm², m²)
- For volume: use cubic units (cm³, m³)
- Round answers appropriately (usually 3 significant figures)
Real-World Applications
- Architecture: Calculating materials needed for buildings
- Packaging: Designing boxes and containers
- Engineering: Building roads, bridges, tunnels
- Medicine: Calculating drug dosages based on volume
- Sports: Designing fields, tracks, equipment
- Cooking: Measuring ingredients and container sizes
Practice Tips
- Always draw a diagram - it helps you visualize the problem
- Label all measurements on your diagram
- Write formulas first before substituting numbers
- Show your working - it helps you find mistakes
- Check if your answer makes sense - is it reasonable?
- Remember π ≈ 3.14 but use the calculator π button for accuracy
Common Mistakes to Avoid
| ❌ Wrong | ✅ Correct |
|---|---|
| Confusing radius and diameter | Remember: diameter = 2 × radius |
| Forgetting to square the radius | Always calculate r² before multiplying by π |
| Using the wrong height | For volume, always use perpendicular height |
| Mixing up 2D and 3D formulas | Check if you need area (flat) or volume (solid) |
| Forgetting units | Always include cm², m³, etc. in your final answer |
Key Vocabulary Review
- Perimeter: Distance around a shape
- Area: Space inside a flat shape
- Circumference: Perimeter of a circle
- Sector: Slice of a circle
- Arc: Curved edge of a sector
- Net: Flat pattern that folds into 3D shape
- Volume: Space inside a solid
- Surface area: Total area of all faces
- Apex: Top point of pyramid/cone
- Slant height: Diagonal distance on cone/pyramid
Quick Reference Guide
• Rectangle: A = b × h
• Triangle: A = ½bh
• Circle: A = πr², C = 2πr
• Sector: A = (θ/360) × πr²
• Cuboid: V = l × w × h
• Cylinder: V = πr²h
• Pyramid: V = (1/3) × base area × h
• Cone: V = (1/3)πr²h
• Sphere: V = (4/3)πr³
🎓 End of Notes 🎓
These notes cover all the essential concepts from sections 7.1-7.3 of your Cambridge textbook.
Practice the exercises in your textbook to master these concepts!
Remember: Mathematics is learned by doing, so work through plenty of examples!
