📐 Perimeter, Area and Volume

Complete Guide for Grade 7 - Cambridge Curriculum

📏 Section 1: Perimeter and Area in Two Dimensions

1.1 Understanding Perimeter

Perimeter is the total distance around the outside of a shape. To find the perimeter, add up the lengths of all sides.

For any polygon:

Perimeter = Sum of all sides

Circle - Circumference

The perimeter of a circle is called circumference.

C = πd

C = 2πr

where π ≈ 3.14 or 22/7

Example 1: Circle Circumference

Find the circumference of a circle with radius 7 cm.

Solution:

C = 2πr
C = 2 × 3.14 × 7
C = 43.96 cm

1.2 Understanding Area

Area is the total space contained within a shape. It is always measured in square units (cm², m², mm²).

Area Formulas for Common Shapes

Square

s ┌─────┐ │ │ s │ │ └─────┘ s

A = s²

Rectangle

b ┌─────┐ │ │ h └─────┘

A = b × h

Triangle

/\ / \ / h \ /______| b

A = (b × h)/2

Circle

___ / \ | r | \___/

A = πr²

Parallelogram and Rhombus

A = base × height

Use perpendicular height!

Kite

A = (d₁ × d₂)/2

where d₁ and d₂ are the diagonals

Trapezium

A = [(a + b) × h]/2

Example 2: Trapezium Area

Parallel sides: 8 cm and 12 cm, Height: 5 cm

Solution:

A = [(8 + 12) × 5]/2
A = [20 × 5]/2
A = 100/2 = 50 cm²

1.3 Complex Shapes

To find the area of complex shapes:

Divide the shape into simpler known shapes
Calculate the area of each part
Add or subtract the areas

1.4 Circles - Arcs and Sectors

Arc Length:

Arc = (θ/360°) × 2πr

Sector Area:

Area = (θ/360°) × πr²

where θ = angle in degrees

Example 3: Sector Area

Circle with radius 12 cm, angle 60°

Solution:

Area = (60/360) × π × 12²
Area = (1/6) × 3.14 × 144
Area = 75.36 cm²

🎲 Section 2: Three-Dimensional Objects

2.1 Common 3D Shapes

Cube

• 6 identical square faces
• All edges equal
• 12 edges, 8 vertices

Cuboid

• 6 rectangular faces
• Like a box
• 12 edges, 8 vertices

Cylinder

• 2 circular ends
• 1 curved surface
• Like a can

Cone

• 1 circular base
• 1 curved surface to point
• Like ice cream cone

Sphere

• Perfectly round
• All points equidistant from center
• Like a ball

Pyramid

• Polygon base
• Triangular faces to apex
• Various types

2.2 Nets

A net is a 2D pattern that can be folded to create a 3D shape. Think of unfolding a cardboard box.

Tip: To identify a 3D shape from its net, count the faces and note their shapes.

📦 Section 3: Surface Area and Volume

3.1 Surface Area

Surface area is the total area of all faces of a 3D object. Measured in square units (cm², m²).

Surface Area Formulas

Shape	Formula
Cube	`SA = 6s²`
Cuboid	`SA = 2(lb + bh + lh)`
Cylinder	`SA = 2πr(r + h)`
Cone (curved)	`SA = πrl`
Cone (total)	`SA = πrl + πr²`
Sphere	`SA = 4πr²`

Example 4: Cuboid Surface Area

Length = 5 cm, Breadth = 3 cm, Height = 2 cm

Solution:

SA = 2(lb + bh + lh)
SA = 2(5×3 + 3×2 + 5×2)
SA = 2(15 + 6 + 10)
SA = 2(31) = 62 cm²

3.2 Volume

Volume is the amount of space inside a 3D object. Measured in cubic units (cm³, m³).

Volume Formulas

Shape	Formula
Cube	`V = s³`
Cuboid	`V = l × b × h`
Any Prism	`V = A × l`
Cylinder	`V = πr²h`
Pyramid	`V = (1/3) × A × h`
Cone	`V = (1/3) × πr²h`
Sphere	`V = (4/3) × πr³`

3.3 Understanding Prisms

A prism is a 3D shape with:

Two identical parallel end faces
Uniform cross-section throughout
Same shape when sliced parallel to ends

Universal Prism Formula:

V = Area of cross-section × length

V = A × l

Example 5: Triangular Prism

Triangle base: 6 cm, height: 4 cm
Prism length: 10 cm

Solution:

Triangle area = (6 × 4)/2 = 12 cm²
Volume = 12 × 10 = 120 cm³

Example 6: Cylinder Volume

Radius: 5 cm, Height: 12 cm

Solution:

V = πr²h
V = 3.14 × 5² × 12
V = 3.14 × 25 × 12
V = 942 cm³

3.4 Pyramids and Cones

Pyramids and cones have volumes that are one-third (1/3) of corresponding prisms.

Remember: Three identical pyramids fit exactly into a prism with the same base and height!

Example 7: Cone Volume

Radius: 4 cm, Height: 9 cm

Solution:

V = (1/3) × πr²h
V = (1/3) × 3.14 × 16 × 9
V = (1/3) × 452.16
V = 150.72 cm³

Example 8: Sphere

Radius: 6 cm

Solution:

Volume = (4/3) × πr³
V = (4/3) × 3.14 × 216
V = 904.32 cm³

Surface Area = 4πr²
SA = 4 × 3.14 × 36
SA = 452.16 cm²

📋 Complete Formula Summary

2D Shapes - Area

Shape	Formula
Square	`s²`
Rectangle	`l × b`
Triangle	`bh/2`
Parallelogram/Rhombus	`bh`
Kite	`(d₁ × d₂)/2`
Trapezium	`[(a + b) × h]/2`
Circle	`πr²`
Sector	`(θ/360°) × πr²`

2D Shapes - Perimeter

Shape	Formula
Any polygon	Sum of all sides
Circle (Circumference)	`2πr` or `πd`
Arc Length	`(θ/360°) × 2πr`