Cambridge Lower Secondary CheckpointStage 9

Geometry and Measure (9Gp)

Mathematics Stage 9 Chapter Notes

What this chapter covers

Geometry and Measure - Position and transformation
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Geometry and Measure (9Gp) notes

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In this chapter

  1. 1Area and Circumference of a Circle
  2. 2Calculating Areas of Compound Shapes
  3. 3Converting Small and Large Metric Units

1. Area and Circumference of a Circle

A circle is a 2D shape where all points on the edge are the same distance from the centre. This distance is the radius (r). The diameter(d) is the distance across the circle through the centre; it's always twice the radius (d = 2r). The circumference (C) is the distance around the circle's edge (its perimeter). The area (A) is the space inside the circle. Both are calculated using the special number Pi (π), which is approximately 3.14159.

C = 2πr

C = πd

A = πr²

Key term

Pi (π): A mathematical constant representing the ratio of a circle's circumference to its diameter, approximately equal to 3.142.

Examiner insight

Examiners look for the correct formula being stated and used. Use the π button on your calculator for accuracy unless told otherwise, and round your final answer to the specified degree of accuracy.

Common pitfall

Students often confuse the formula for area (πr²) with the formula for circumference (2πr). Remember that area involves 'squared' units, so the formula with r² is for area.

Worked example 14 marks

A circle has a radius of 5 cm. Calculate:a) its circumference,b) its area. Give your answers to 3 significant figures.

  1. 1

    a) Circumference: Identify the correct formula. C = 2πr.

  2. 2

    Substitute the radius: C = 2 × π × 5 cm.

  3. 3

    Calculate: C = 10π cm ≈ 31.4159... cm.

  4. 4

    Round to 3 significant figures: C = 31.4 cm.

  5. 5

    b) Area: Identify the correct formula. A = πr².

  6. 6

    Substitute the radius: A = π × (5 cm)² = π × 25 cm².

  7. 7

    Calculate: A = 25π cm² ≈ 78.5398... cm².

  8. 8

    Round to 3 significant figures: A = 78.5 cm².

Worked example 23 marks

The area of a circular pizza is 400 cm². What is its diameter? Give your answer to 1 decimal place.

  1. 1

    Start with the area formula: A = πr².

  2. 2

    Substitute the known area: 400 = πr².

  3. 3

    Rearrange to find the radius squared: r² = 400 / π.

  4. 4

    Calculate r²: r² ≈ 127.3239...

  5. 5

    Square root to find the radius: r = √(127.3239...) ≈ 11.2837... cm.

  6. 6

    The question asks for the diameter. Remember d = 2r.

  7. 7

    Calculate the diameter: d = 2 × 11.2837... ≈ 22.5675... cm.

  8. 8

    Round to 1 decimal place: d = 22.6 cm.

Recap

  • The circumference is the distance around a circle: C = 2πr or C = πd.
  • The area is the space inside a circle: A = πr².
  • The diameter is double the radius (d = 2r).
  • Use the π button on your calculator for the most accurate result unless told otherwise.
  • Always check if the question gives you the radius or the diameter before you start.

Quick check

  1. What is the area of a circle with a diameter of 20 m? Leave your answer in terms of π.2 marks

2. Calculating Areas of Compound Shapes

Compound shapes are figures made by joining two or more simple shapes. To find the area of a compound shape, you must break it down into basic shapes you recognise, like rectangles, triangles, and circles (or parts of circles). Calculate the area of each part separately and then add them together. Sometimes, it's easier to calculate the area of a larger, simpler shape and subtract the area of a missing piece.

Area of Rectangle = length × width

Area of Triangle = ½ × base × height

Area of Trapezium = ½ × (a+b) × height

Area of Circle = πr²

Key term

Compound Shape: A 2D figure made from two or more basic geometric shapes combined.

Examiner insight

Examiners award marks for showing a clear method. Draw lines on the diagram to show how you've split the shape and label the areas (e.g., A1, A2) you are calculating.

Common pitfall

When a shape is split, students sometimes use an overall dimension for a smaller part. For example, using the full width of the compound shape for the base of a smaller triangle within it. Always recalculate the dimensions for each individual part.

Worked example 14 marks

Calculate the area of the shape below, which consists of a rectangle and a semicircle. The rectangle has a length of 10 cm and a width of 8 cm, with the semicircle attached to one of the 8 cm sides. Give your answer to 3 significant figures.

  1. 1

    Step 1: Decompose the shape into a rectangle (Part A) and a semicircle (Part B).

  2. 2

    Step 2: Calculate the area of the rectangle (A). Area_A = length × width = 10 cm × 8 cm = 80 cm².

  3. 3

    Step 3: Find the dimensions for the semicircle (B). The diameter of the semicircle is the side of the rectangle, so d = 8 cm. The radius is half the diameter, so r = 4 cm.

  4. 4

    Step 4: Calculate the area of a full circle with this radius: Area_full_circle = πr² = π × (4 cm)² = 16π cm².

  5. 5

    Step 5: The area of the semicircle (B) is half the area of the full circle. Area_B = ½ × 16π cm² = 8π cm² ≈ 25.1327... cm².

  6. 6

    Step 6: Add the areas together. Total Area = Area_A + Area_B = 80 + 25.1327... = 105.1327... cm².

  7. 7

    Step 7: Round to 3 significant figures. Total Area = 105 cm².

Worked example 23 marks

The diagram shows a circular piece of card with a radius of 6 cm, from which a square hole of side length 4 cm is cut from the centre. Calculate the area of the remaining card. Give your answer to 1 decimal place.

  1. 1

    Step 1: Identify the strategy. This is a subtraction problem: Area of Card = Area of Circle - Area of Square.

  2. 2

    Step 2: Calculate the area of the large circle. The radius is 6 cm. Area_Circle = πr² = π × (6 cm)² = 36π cm².

  3. 3

    Calculate the value: Area_Circle ≈ 113.097... cm².

  4. 4

    Step 3: Calculate the area of the square hole. The side length is 4 cm. Area_Square = side × side = 4 cm × 4 cm = 16 cm².

  5. 5

    Step 4: Subtract the area of the square from the area of the circle. Shaded Area = 113.097... - 16 = 97.097... cm².

  6. 6

    Step 5: Round the final answer to 1 decimal place. Shaded Area = 97.1 cm².

Recap

  • Break down compound shapes into simpler shapes like rectangles, triangles, and circles.
  • Calculate the area of each simple shape individually.
  • Add the areas together if shapes are combined.
  • Subtract areas if a shape has a piece cut out of it.
  • Be careful to find the correct dimensions for each individual part.

Quick check

  1. An L-shaped room has a long side of 10m, a short side of 4m, and a width of 3m everywhere. What is its total area?2 marks

3. Converting Small and Large Metric Units

In science and maths, we use standard metric units for measurement. You need to be able to convert between them for length (metres), mass (grams), and capacity (litres). For very large or small quantities, we often use standard form. When converting units of area (e.g., m² to cm²) or volume (e.g., m³ to cm³), you must apply the conversion factor twice or three times, respectively.

1 km = 1000 m

1 m = 100 cm

1 cm = 10 mm

1 kg = 1000 g

1 tonne = 1000 kg

1 litre = 1000 ml

1 m² = 10,000 cm²

1 cm² = 100 mm²

Key term

Standard Form: A way of writing very large or very small numbers compactly, in the form A × 10^n, where 1 ≤ A < 10 and n is an integer.

Examiner insight

Marks are often awarded for the correct conversion factor, even if the final calculation is wrong. Show your conversion steps clearly, for example, 'To convert 5 m² to cm², I calculate 5 × (100)²'.

Common pitfall

Forgetting to square or cube the conversion factor for area and volume. For example, to convert 2 m² to cm², many students incorrectly multiply by 100 instead of 100², giving 200 cm² instead of the correct 20,000 cm².

Fun fact

A nanometre (nm) is one billionth of a metre (10⁻⁹ m). A single human hair is about 80,000 to 100,000 nanometres wide.

Worked example 12 marks

A rectangular field measures 0.5 km by 200 m. Calculate its area in square metres (m²).

  1. 1

    Step 1: Ensure all units are consistent before calculating. Let's convert everything to metres.

  2. 2

    The width is already in metres: 200 m.

  3. 3

    Convert the length from km to m. 1 km = 1000 m, so 0.5 km = 0.5 × 1000 = 500 m.

  4. 4

    Step 2: Calculate the area using the converted units. Area = length × width = 500 m × 200 m.

  5. 5

    Step 3: Perform the calculation. Area = 100,000 m².

Worked example 22 marks

Convert an area of 3.5 m² into square centimetres (cm²).

  1. 1

    Step 1: Identify the linear conversion factor. 1 m = 100 cm.

  2. 2

    Step 2: Since we are converting area (a 2D measure), we must square the linear conversion factor. The area conversion factor is (100)² = 10,000.

  3. 3

    Step 3: Multiply the original area by the area conversion factor. Area in cm² = 3.5 × 10,000.

  4. 4

    Step 4: Calculate the final answer. Area = 35,000 cm².

Worked example 32 marks

The mass of a dust particle is 0.000000753 kg. Write this mass in grams and in standard form.

  1. 1

    Step 1: Convert the mass from kg to g. 1 kg = 1000 g.

  2. 2

    Mass in g = 0.000000753 × 1000 = 0.000753 g.

  3. 3

    Step 2: Write this mass in grams in standard form.

  4. 4

    Standard form is A × 10^n where 1 ≤ A < 10. We need to move the decimal point 4 places to the right to get 7.53.

  5. 5

    Since we moved the decimal point to the right for a number less than 1, the power will be negative.

  6. 6

    Mass = 7.53 × 10⁻⁴ g.

Recap

  • To convert from a larger unit to a smaller unit, you multiply.
  • To convert from a smaller unit to a larger unit, you divide.
  • For area conversions (e.g., m² to cm²), square the length conversion factor.
  • For volume conversions (e.g., m³ to cm³), cube the length conversion factor.
  • Standard form is a shorthand for writing very large or very small numbers.

Quick check

  1. How many millilitres are in 2.5 litres?1 mark
  2. A paving slab is a square with sides of 50 cm. What is its area in m²?2 marks

End-of-chapter exercise

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

  1. A circle has a diameter of 14 cm. Calculate its area. Use π = 22/7.2 marks
  2. An L-shaped polygon has a total height of 8 cm and a total width of 6 cm. The width of each 'arm' of the L is 2 cm. Calculate the total area of the shape.3 marks
  3. A running track is formed by a rectangle of length 100m and width 50m, with a semicircle at each end. Calculate the total distance around the outside of the track (the perimeter). Give your answer to the nearest metre.4 marks
  4. A circular lawn has a radius of 10m. A circular flowerbed with a radius of 3m is cut from the centre of the lawn. What is the area of the remaining grass? Give your answer to 3 significant figures.3 marks
  5. A farmer's field is a trapezium with parallel sides of length 300m and 500m. The perpendicular distance between the parallel sides is 250m. Calculate the area of the field in square kilometres (km²).4 marks
  6. The circumference of a bicycle wheel is 215 cm. What is the radius of the wheel? Give your answer correct to the nearest centimetre.3 marks
  7. A sheet of A4 paper measures 29.7 cm by 21.0 cm. What is its area in square millimetres (mm²)?3 marks
  8. A window is made from a square of side 60 cm with a semicircle on top. A 2 cm wide wooden frame runs around the entire outer edge of the window. Calculate the area of the glass. Give your answer to 3 significant figures.5 marks
  9. A decorative patio is shaped like a sector of a circle with a radius of 4m and an angle of 90°. Find the perimeter and area of the patio. Give your answers in terms of π.4 marks
  10. The Earth has a radius of approximately 6,371 km. The surface area of a sphere is given by the formula A = 4πr². Calculate the surface area of the Earth in square metres, giving your answer in standard form correct to 3 significant figures.4 marks

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