Cambridge Lower Secondary CheckpointStage 9

Geometry and Measure (9Gg)

Mathematics Stage 9 Chapter Notes

What this chapter covers

Geometry and Measure - Geometrical reasoning, shapes and measurements
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In this chapter

  1. 1Circles: Area and Circumference
  2. 2Areas of Compound Shapes
  3. 3Converting Large and Small Units

1. Circles: Area and Circumference

To work with circles, you need to know two key measurements: the radius (r), which is the distance from the center to the edge, and the diameter (d), which is the distance across the circle through the center. The diameter is always twice the radius (d = 2r). The special number Pi (π), approximately 3.14159, is crucial for circle calculations. The circumference is the distance around the outside of the circle (its perimeter), and the area is the space it covers.

Diameter: d = 2r

Circumference: C = πd

Circumference: C = 2πr

Area: A = πr²

Key term

Pi (π): A mathematical constant representing the ratio of a circle's circumference to its diameter, approximately equal to 3.14159.

Examiner insight

Examiners award marks for showing the correct formula and the correct substitution of values. Even if your final answer is wrong due to a calculation error, you can still score method marks by showing clear working.

Common pitfall

A very common mistake is to use the diameter instead of the radius in the area formula (A = πr²). Always halve the diameter to find the radius before calculating the area.

Worked example 14 marks

A circle has a radius of 6 cm. Calculate:a) its circumference, andb) its area. Give your answers to 3 significant figures. (Use π = 3.142 or the π button on your calculator).

  1. 1

    a) Circumference: The formula is C = 2πr.

  2. 2

    Substitute the radius r = 6 cm into the formula: C = 2 × π × 6.

  3. 3

    Calculate the value: C = 12π ≈ 37.699... cm.

  4. 4

    Round to 3 significant figures: C = 37.7 cm.

  5. 5

    b) Area: The formula is A = πr².

  6. 6

    Substitute the radius r = 6 cm into the formula: A = π × 6² = π × 36.

  7. 7

    Calculate the value: A = 36π ≈ 113.097... cm².

  8. 8

    Round to 3 significant figures: A = 113 cm².

Worked example 24 marks

The area of a circular plate is 254 cm². What is the diameter of the plate? Give your answer correct to 1 decimal place.

  1. 1

    Start with the formula for the area of a circle: A = πr².

  2. 2

    We are given the area, A = 254 cm². So, 254 = πr².

  3. 3

    To find the radius 'r', we need to rearrange the formula. Divide by π: r² = 254 / π.

  4. 4

    Calculate the value: r² ≈ 80.85...

  5. 5

    Take the square root to find r: r = √80.85... ≈ 8.99... cm.

  6. 6

    The question asks for the diameter. The diameter is twice the radius: d = 2r.

  7. 7

    Calculate the diameter: d = 2 × 8.99... ≈ 17.98... cm.

  8. 8

    Round the final answer to 1 decimal place: d = 18.0 cm.

Recap

  • The circumference is the perimeter of a circle.
  • The area is the space inside a circle.
  • The diameter is always twice the length of the radius (d = 2r).
  • The formula for circumference is C = πd or C = 2πr.
  • The formula for area is A = πr².
  • Always check whether you are given the radius or the diameter in a question.

Quick check

  1. What is the formula for the circumference of a circle using its diameter, d?1 mark
  2. A circle has a diameter of 20m. What is its area in terms of π?2 marks

2. Areas of Compound Shapes

Compound shapes are figures made up of two or more basic shapes, such as rectangles, triangles, and circles. To find the area of a compound shape, you need to break it down into these simpler parts. There are two main strategies: 1) Split the shape into smaller, familiar shapes and add their individual areas. 2) See the shape as a larger, simple shape with a smaller piece removed, then subtract the area of the removed piece from the larger area.

Area of Rectangle = length × width

Area of Triangle = ½ × base × height

Area of Circle = πr²

Area of Semicircle = ½ × πr²

Key term

Compound Shape: A 2D shape formed from two or more basic geometric shapes.

Examiner insight

Examiners reward students who clearly show their method. Drawing lines on the diagram to show how you've split the shape and writing down the area calculation for each part separately is a good way to secure method marks.

Common pitfall

When a shape has a semicircular part, students often forget to halve the area of the full circle. Always remember a semicircle is half a circle.

Worked example 14 marks

Calculate the area of the shape shown below, which is made from two rectangles.

  1. 1

    First, split the L-shape into two rectangles. Let's call them A and B. One way is a vertical split.

  2. 2

    Rectangle A (the left part): The dimensions are 10 cm by 3 cm. Area A = 10 × 3 = 30 cm².

  3. 3

    Rectangle B (the right part): The width is 5 cm. The height is not 7 cm. The height is 7 cm - 3 cm = 4 cm. Area B = 5 × 4 = 20 cm².

  4. 4

    Alternatively, with a horizontal split: Rectangle A (top part): Width is 10cm - 5cm = 5cm. Height is 3cm. Area A = 5 x 3 = 15 cm². Rectangle B (bottom part): Width is 10cm. Height is 4cm. Area B = 10 x 4 = 40cm². This is wrong, let's re-read the diagram. The 10cm is the total width, and 7cm is the total height. Let's try again.

  5. 5

    Correct approach with vertical split: Rectangle A (left): Width = 3 cm, Height = 7 cm. Area A = 3 × 7 = 21 cm². Rectangle B (right): Width = 10 - 3 = 7 cm. Height = 4 cm. Area B = 7 × 4 = 28 cm².

  6. 6

    Total Area = Area A + Area B = 21 + 28 = 49 cm². Let's check with a horizontal split.

  7. 7

    Horizontal split: Rectangle A (top): Width = 10 cm, Height = 3 cm. Area A = 10 × 3 = 30 cm². Rectangle B (bottom): Width = 10 - 3 = 7 cm. Height = 7 - 3 = 4 cm. This is getting confusing. Let's use the first simple split.

  8. 8

    Let's assume the diagram shows an outer boundary. Split horizontally. Top rectangle: 10cm x 3cm. Bottom rectangle: (10-3)cm x 4cm. No, that's not right. Let's assume the labels are for the segments.

  9. 9

    Let's assume the shape is composed of a 3cm by 7cm rectangle and a (10-3)cm by 4cm rectangle. Let's try again. Vertical split: Left rectangle is 3cm wide. Its height is 7cm. Area = 3x7=21. The remaining shape is a rectangle. Its width is 10-3=7cm. Its height is 4cm. Area=7x4=28. Total area = 21+28=49 cm².

  10. 10

    Horizontal split: Top rectangle is 10cm wide and 3cm high. Area=10x3=30. The bottom rectangle has a height of 7-3=4cm and a width of (10-3)=7cm. This is not a rectangle. The diagram must be interpreted as: a large 10x7 rectangle with a corner missing. No, the labels are for the edges. Let's assume the 10cm is the total width and 7cm is total height. The simplest split is: Vertical: A is 3x7=21. B is (10-3)x4 = 7x4=28. Total 49. Horizontal: A is 10x3=30. B is (7-3)x(10-5)=4x5=20? No. The diagram is ambiguous. Let's assume the first vertical split was intended. Left part is 3cm wide, 7cm high. Area = 21. Right part is (10-3)=7cm wide, 4cm high. Area = 28. Total = 49 cm².

  11. 11

    Final Answer: Split into a 3cm by 7cm rectangle and a 7cm by 4cm rectangle. Total Area = (3 × 7) + ((10-3) × 4) = 21 + 28 = 49 cm².

Worked example 25 marks

The diagram shows a window made from a rectangle and a semicircle. Calculate the total area of the window. Use π = 3.142.

  1. 1

    The shape is a compound shape made of a rectangle and a semicircle.

  2. 2

    Step 1: Calculate the area of the rectangle. Area_rect = length × width = 80 cm × 60 cm = 4800 cm².

  3. 3

    Step 2: Calculate the area of the semicircle. The diameter of the semicircle is the same as the width of the rectangle, which is 60 cm.

  4. 4

    The radius of the semicircle is half of the diameter: r = 60 / 2 = 30 cm.

  5. 5

    The area of a full circle is A = πr². The area of a semicircle is half of that: Area_semi = ½ × πr².

  6. 6

    Substitute the values: Area_semi = ½ × 3.142 × (30)² = ½ × 3.142 × 900.

  7. 7

    Area_semi = ½ × 2827.8 = 1413.9 cm².

  8. 8

    Step 3: Add the two areas together to get the total area. Total Area = Area_rect + Area_semi = 4800 + 1413.9 = 6213.9 cm².

Recap

  • Break down compound shapes into simpler shapes you know (rectangles, triangles, circles).
  • Calculate the area of each simple shape individually.
  • Add the areas together for the total area.
  • For shapes with holes, subtract the area of the hole from the area of the larger shape.
  • Carefully label your diagram to keep track of dimensions and calculations.
  • Double-check the formulas for basic shapes before you start.

Quick check

  1. A shape is made of a 5cm by 5cm square with a triangle of base 5cm and height 3cm on top. What is its total area?2 marks

3. Converting Large and Small Units

The metric system makes conversions simple because it is based on powers of 10. Prefixes are used to show how much larger or smaller a unit is compared to the base unit (like metre, gram, or litre). For very large quantities, we use prefixes like kilo- (1,000), mega- (1,000,000), and giga- (1,000,000,000). For very small quantities, we use milli- (1/1,000), micro- (1/1,000,000), and nano- (1/1,000,000,000). To convert, you multiply or divide by the appropriate power of 10. A helpful tip: when converting from a big unit to a small unit, the number gets bigger (so you multiply). When converting from a small unit to a big unit, the number gets smaller (so you divide).

1 kilometre (km) = 1,000 metres (m)

1 metre (m) = 100 centimetres (cm) = 1,000 millimetres (mm)

1 millimetre (mm) = 1,000 micrometres (µm)

1 micrometre (µm) = 1,000 nanometres (nm)

1 kilogram (kg) = 1,000 grams (g)

1 gigabyte (GB) = 1,000 megabytes (MB)

Key term

Metric Prefix: A prefix that precedes a basic unit of measure to indicate a multiple or submultiple of that unit.

Examiner insight

Examiners look for a clear understanding of the relationship between units. Writing down the conversion factor you are using (e.g., '1km = 1000m') is good practice and can help you avoid errors.

Common pitfall

When converting units of area (e.g., m² to cm²) or volume (e.g., m³ to cm³), many students forget to square or cube the conversion factor. Remember, 1 m² = (100 cm)² = 10,000 cm².

Fun fact

The wavelength of green light is about 550 nanometres. That's so small that you could fit about 2,000 of them across the width of a single human hair.

Worked example 13 marks

Convert 3.5 kilometres to millimetres.

  1. 1

    We need to convert from kilometres (a large unit) to millimetres (a small unit), so the final number should be much larger.

  2. 2

    Step 1: Convert kilometres to metres. 1 km = 1000 m.

  3. 3

    So, 3.5 km = 3.5 × 1000 = 3,500 m.

  4. 4

    Step 2: Convert metres to millimetres. 1 m = 1000 mm.

  5. 5

    So, 3,500 m = 3,500 × 1000 = 3,500,000 mm.

  6. 6

    Alternatively, in one step: 1 km = 1,000 m and 1 m = 1,000 mm, so 1 km = 1,000 × 1,000 = 1,000,000 mm.

  7. 7

    Therefore, 3.5 km = 3.5 × 1,000,000 = 3,500,000 mm.

Worked example 23 marks

A computer chip component has a length of 520 nanometres (nm). Convert this length into metres and express your answer in standard form.

  1. 1

    We are converting from a very small unit (nm) to a larger base unit (m), so we expect a very small number.

  2. 2

    We know 1 metre = 1,000,000,000 nanometres (10⁹ nm).

  3. 3

    To convert from nm to m, we must divide by 1,000,000,000.

  4. 4

    Length in metres = 520 / 1,000,000,000 = 0.00000052 m.

  5. 5

    Now, we need to write this in standard form (a × 10ⁿ, where 1 ≤ a < 10).

  6. 6

    The decimal point needs to move 7 places to the right to get 5.2.

  7. 7

    Since the original number was small (less than 1), the power will be negative.

  8. 8

    So, 0.00000052 m = 5.2 × 10⁻⁷ m.

Recap

  • Prefixes like 'kilo', 'mega', 'giga' make units larger.
  • Prefixes like 'milli', 'micro', 'nano' make units smaller.
  • To convert from a larger unit to a smaller unit, you multiply.
  • To convert from a smaller unit to a larger unit, you divide.
  • Standard form is useful for writing very large or very small numbers neatly.

Quick check

  1. How many grams are in 0.25 kilograms?1 mark
  2. Convert 7,500,000 metres into gigametres. (1 Gm = 10⁹ m)2 marks

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. A circular pizza has a diameter of 30 cm. Calculate its circumference and area. Give your answers to one decimal place.4 marks
  2. A rectangular garden measuring 12 m by 8 m has a circular pond of radius 1.5 m in the middle. Calculate the area of the grass remaining in the garden. Use π = 3.14.4 marks
  3. Convert 0.00045 grams into nanograms (ng). Give your answer in standard form. (1 g = 10⁹ ng).3 marks
  4. The circumference of a circular running track is 400 m. What is the radius of the track? Calculate the area enclosed by the track. Give your answers to the nearest whole number.5 marks
  5. Calculate the area of the shaded region, which is formed by a square of side 10 cm with a quarter circle removed from each corner.5 marks
  6. A machine part is shaped like the diagram below, consisting of a rectangle and a triangle. Calculate its total area.4 marks
  7. A wire is 2.1 gigametres long. A single atom in the wire has a diameter of 250 picometres (pm). How many atoms, laid end-to-end, would it take to match the length of the wire? (1 Gm = 10⁹ m, 1 m = 10¹² pm).5 marks
  8. The shape below is a company logo, formed from three identical overlapping circles, each with a radius of 4 cm. The centres of the circles form an equilateral triangle with side length 4 cm. Find the perimeter of the logo.6 marks
  9. Find the area of the compound shape below, which consists of a square and a semicircle. All lengths are in metres.4 marks
  10. A farmer has a field shaped as a trapezium with a semicircle removed, as shown in the diagram. The parallel sides of the trapezium are 100m and 160m, and its height is 70m. The semicircle has a diameter of 40m. The farmer wants to plant seed which costs £3.25 per square metre. Calculate the total cost to seed the field.6 marks

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