Scale Drawings, Coordinates and Angles

This lesson covers DfE content statements L2.18, L2.19 and L2.22 — using and creating scale drawings and maps, using coordinates in all four quadrants, and calculating angles using angle facts.

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Scale Drawings

A scale drawing is a diagram where every measurement is in the same proportion to the actual object. The scale tells you the relationship between the drawing and real life.

Common Scale Formats

Format	Meaning	Example
1 : n	1 unit on drawing = n units in real life	1 : 50 means 1 cm = 50 cm (0.5 m)
1 cm = x m	Each centimetre represents x metres	1 cm = 2 m
Ratio bar	A graphical bar on a map	—

Worked Example — Floor Plan

Scenario: An architect's floor plan has a scale of 1 : 50. A room measures 8 cm × 6 cm on the plan. What are the actual dimensions?

Length = 8 × 50 = 400 cm = 4 m Width = 6 × 50 = 300 cm = 3 m

Worked Example — Drawing to Scale

Scenario: You need to draw a garden that is 12 m × 8 m at a scale of 1 : 200. What size will it be on paper?

Length = 12 m ÷ 200 = 0.06 m = 6 cm Width = 8 m ÷ 200 = 0.04 m = 4 cm

Worked Example — Map Distance

Scenario: A map has a scale of 1 : 25,000. Two villages are 14 cm apart on the map. What is the actual distance?

14 × 25,000 = 350,000 cm = 3,500 m = 3.5 km

Exam Tip: When working with scales, be very careful with unit conversions. It helps to convert everything to centimetres first, then convert the final answer to metres or kilometres at the end.

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Reading and Creating Scale Drawings

graph LR
    A["Measure the<br/>real object"] --> B["Choose an<br/>appropriate scale"]
    B --> C["Divide real<br/>measurements<br/>by the scale factor"]
    C --> D["Draw accurately<br/>using a ruler<br/>and protractor"]

Choosing a Scale

The scale should:

• Make the drawing fit on your paper
• Use numbers that are easy to work with
• Be clearly stated on the drawing

Scenario: A room is 6 m × 4.5 m. Your paper is A4 (roughly 30 cm × 21 cm). What scale would work?

At 1 : 20: room = 30 cm × 22.5 cm (too big for A4) At 1 : 25: room = 24 cm × 18 cm (fits well) ✓

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Coordinates

Coordinates describe the position of a point on a grid. The format is (x, y) where x is the horizontal distance and y is the vertical distance from the origin (0, 0).

Four Quadrants

The x and y axes divide the grid into four quadrants:

Quadrant	x value	y value	Example
First (top right)	Positive	Positive	(3, 4)
Second (top left)	Negative	Positive	(−2, 5)
Third (bottom left)	Negative	Negative	(−3, −1)
Fourth (bottom right)	Positive	Negative	(4, −2)

Worked Example — Plotting Points

Scenario: A surveyor maps four corners of a building at coordinates: A(2, 3), B(8, 3), C(8, 7), D(2, 7). What shape is the building and what are its dimensions?

AB is horizontal: 8 − 2 = 6 units BC is vertical: 7 − 3 = 4 units The shape is a rectangle, 6 units × 4 units.

Finding the Midpoint

The midpoint of a line segment is found by averaging the x-coordinates and averaging the y-coordinates.

Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Scenario: Find the midpoint of the line from (2, 5) to (8, 11).

Midpoint = ((2 + 8)/2, (5 + 11)/2) = (5, 8)

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Angles

An angle is the amount of turn between two lines meeting at a point, measured in degrees (°).

Types of Angles

Type	Size	Description
Acute	Less than 90°	Sharp angle
Right angle	Exactly 90°	Quarter turn; marked with a square
Obtuse	Between 90° and 180°	Wide angle
Reflex	Between 180° and 360°	More than a straight line
Straight line	Exactly 180°	Half turn
Full turn	Exactly 360°	Complete rotation

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Angle Facts

These rules allow you to calculate missing angles without measuring.

1. Angles on a Straight Line = 180°

If angles sit on a straight line, they add up to 180°.

Scenario: A ramp makes an angle of 35° with the ground. What is the angle on the other side of the ramp?

180° − 35° = 145°

2. Angles Around a Point = 360°

All angles meeting at a single point add up to 360°.

Scenario: A pie chart has three sectors measuring 120°, 85° and an unknown angle. Find the missing angle.