Length and Pythagoras Rule – Basic 6 Mathematics Lesson Note (Second Term)

Units of Length

Common Units of Length

• Millimeter (mm) – smallest unit
• Centimeter (cm) – 10 mm = 1 cm
• Meter (m) – 100 cm = 1 m
• Kilometer (km) – 1000 m = 1 km

Converting Between Units

Examples:

• 5 m = 500 cm
• 3.5 km = 3,500 m
• 25 mm = 2.5 cm

Exercise A – Unit Conversion

Convert these measurements:

1. 8 m = ______ cm
2. 1.2 km = ______ m
3. 450 cm = ______ m
4. 65 mm = ______ cm
5. 2.5 m = ______ mm
6. 7,500 m = ______ km
7. 38 cm = ______ mm
8. 0.75 km = ______ m

Measuring Length

Using Rulers and Measuring Tapes

Tips for Accurate Measurement:

• Start measuring from the 0 mark
• Keep the ruler straight
• Read at eye level
• Round to the nearest unit when necessary

Estimating Length

Common Reference Points:

• Width of thumb ≈ 2 cm
• Length of foot ≈ 25 cm
• Height of door ≈ 2 m
• Length of classroom ≈ 8 m

Exercise B – Estimation and Measurement

Estimate, then measure if possible:

1. Estimate the length of your desk: ______ cm
2. Estimate the width of your exercise book: ______ cm
3. Estimate the height of your chair: ______ cm
4. Estimate the length of your pencil: ______ cm
5. Which is longer: Your arm span or your height? ______

Perimeter

What is Perimeter?

Perimeter is the distance around the outside of a shape.

Finding Perimeter

Rectangle: Perimeter = 2(length + width) Square: Perimeter = 4 × side Triangle: Perimeter = side 1 + side 2 + side 3

Examples:

• Rectangle: length = 8 cm, width = 5 cm Perimeter = 2(8 + 5) = 2 × 13 = 26 cm
• Square: side = 6 cm Perimeter = 4 × 6 = 24 cm

Exercise C – Perimeter Calculations

Find the perimeter of these shapes:

1. Rectangle: length = 12 cm, width = 7 cm Perimeter: ______ cm
2. Square: side = 9 cm Perimeter: ______ cm
3. Triangle: sides = 8 cm, 6 cm, 10 cm Perimeter: ______ cm
4. Rectangle: length = 15 m, width = 8 m Perimeter: ______ m
5. Regular pentagon: each side = 4 cm Perimeter: ______ cm

Introduction to Pythagoras Rule

What is Pythagoras Rule?

In a right-angled triangle, the square of the longest side (hypotenuse) equals the sum of squares of the other two sides.

Formula: a² + b² = c²

Where:

• a and b are the shorter sides
• c is the hypotenuse (longest side)

Parts of a Right-Angled Triangle

Right angle: 90° angle (shown with small square) Hypotenuse: Side opposite the right angle (longest side) Other sides: The two sides forming the right angle

Simple Example:

Triangle with sides 3 cm, 4 cm, and 5 cm Check: 3² + 4² = 9 + 16 = 25 = 5² ✓

Exercise D – Identifying Right-Angled Triangles

Check if these form right-angled triangles using Pythagoras Rule:

1. Sides: 5 cm, 12 cm, 13 cm Check: 5² + 12² = ______ and 13² = ______ Right-angled? ______
2. Sides: 6 cm, 8 cm, 10 cm Check: 6² + 8² = ______ and 10² = ______ Right-angled? ______
3. Sides: 7 cm, 24 cm, 25 cm Check: 7² + 24² = ______ and 25² = ______ Right-angled? ______

Using Pythagoras Rule

Finding the Hypotenuse

When you know the two shorter sides:

Example: Sides are 6 cm and 8 cm. Find the hypotenuse. c² = a² + b² c² = 6² + 8² = 36 + 64 = 100 c = √100 = 10 cm

Finding a Shorter Side

When you know the hypotenuse and one side:

Example: Hypotenuse = 13 cm, one side = 5 cm. Find the other side. a² + b² = c² 5² + b² = 13² 25 + b² = 169 b² = 169 – 25 = 144 b = √144 = 12 cm

Exercise E – Using Pythagoras Rule

Find the missing sides:

1. Right triangle: sides = 9 cm and 12 cm Hypotenuse: ______ cm
2. Right triangle: hypotenuse = 17 cm, one side = 8 cm Other side: ______ cm
3. Right triangle: sides = 15 cm and 20 cm Hypotenuse: ______ cm
4. Right triangle: hypotenuse = 26 cm, one side = 10 cm Other side: ______ cm
5. Right triangle: sides = 21 cm and 28 cm Hypotenuse: ______ cm

Real Life Applications

Building and Construction

Example: A ladder problem A 5-meter ladder leans against a wall. The bottom of the ladder is 3 meters from the wall. How high up the wall does the ladder reach?

Solution: Wall height² + ground distance² = ladder length² h² + 3² = 5² h² + 9 = 25 h² = 16 h = 4 meters

Exercise F – Real Life Problems

Solve these practical problems:

1. Garden Problem: A rectangular garden is 8 meters long and 6 meters wide. What is the diagonal distance across the garden? Answer: ______ meters
2. Television Screen: A TV screen is 24 cm wide and 18 cm tall. What is the diagonal measurement of the screen? Answer: ______ cm
3. Roof Construction: A roof beam is 10 meters long. One end is 6 meters horizontally from the wall. How high is the other end above the starting point? Answer: ______ meters

Exercise G – Measurement in School

Apply Pythagoras Rule to school situations:

1. Playground Diagonal: A rectangular playground is 30 meters long and 40 meters wide. Students want to run diagonally across it. How far is the diagonal? Answer: ______ meters
2. Classroom Door: A classroom door is 80 cm wide and 200 cm tall. What is the diagonal measurement? Answer: ______ cm (round to nearest cm)
3. Flag Pole: A flag pole casts a shadow 12 meters long. The top of the pole is 16 meters from the tip of the shadow. How tall is the flag pole? Answer: ______ meters

Exercise H – Sports Applications

Use Pythagoras in sports scenarios:

1. Football Field: A football goal post is 7.32 meters wide and 2.44 meters high. What is the diagonal distance from one bottom corner to the opposite top corner? Answer: ______ meters (round to 2 decimal places)
2. Basketball Court: From the center of a basketball court to the corner is 15 meters. If the court is 28 meters long, how wide is it? Answer: ______ meters

Special Right-Angled Triangles

Common Pythagorean Triples

These are sets of three whole numbers that form right-angled triangles:

• 3, 4, 5 (and multiples: 6, 8, 10 or 9, 12, 15)
• 5, 12, 13 (and multiples: 10, 24, 26)
• 8, 15, 17
• 7, 24, 25

Exercise I – Pythagorean Triples

Complete these Pythagorean triples:

1. 3, 4, ______
2. 5, ______, 13
3. ______, 15, 17
4. 7, ______, 25
5. 9, 12, ______ (multiple of 3, 4, 5)

Exercise J – Problem Solving

Solve these challenge problems:

1. Two Measurements: A rectangular field has a diagonal of 25 meters and one side of 15 meters. What is the length of the other side? Answer: ______ meters
2. Square Diagonal: A square has sides of 10 cm. What is the length of its diagonal? Answer: ______ cm (round to 1 decimal place)
3. Isosceles Triangle: An isosceles triangle has two equal sides of 13 cm each and a base of 10 cm. What is the height of the triangle? Answer: ______ cm

Answer Key

Exercise A:

1. 800 cm, 2. 1,200 m, 3. 4.5 m, 4. 6.5 cm, 5. 2,500 mm, 6. 7.5 km, 7. 380 mm, 8. 750 m

Exercise C:

1. 38 cm, 2. 36 cm, 3. 24 cm, 4. 46 m, 5. 20 cm

Exercise D:

1. 25 + 144 = 169, 169, Yes
2. 36 + 64 = 100, 100, Yes
3. 49 + 576 = 625, 625, Yes

Exercise E:

1. 15 cm, 2. 15 cm, 3. 25 cm, 4. 24 cm, 5. 35 cm

Exercise F:

1. 10 meters, 2. 30 cm, 3. 8 meters

Exercise G:

1. 50 meters, 2. 215 cm, 3. Unable to solve (need more information)

Exercise H:

1. 7.71 meters, 2. Unable to solve (court width would be found differently)

Exercise I:

1. 5, 2. 12, 3. 8, 4. 24, 5. 15

Exercise J:

1. 20 meters, 2. 14.1 cm, 3. 12 cm