Cambridge Lower Secondary CheckpointStage 8

Geometry and Measure (8Gp)

Mathematics Stage 8 Chapter Notes

What this chapter covers

Geometry and Measure - Position and transformation
ShareWhatsAppPost
Geometry and Measure (8Gp) notes

Unable to load PDF

The notes viewer could not load. Please refresh the page.

Read online free. Download a watermarked copy with a free account.

Read the notes

The full Geometry and Measure (8Gp) notes as text: skim, search, and jump between subtopics.

~10 min read

In this chapter

  1. 1Converting Between Miles and Kilometres
  2. 2Area of Parallelograms and Trapezia
  3. 3Volume of Triangular Prisms
  4. 4Surface Area of 3D Shapes

1. Converting Between Miles and Kilometres

Distances can be measured in different units. In the UK and USA, miles are common, while most other countries use kilometres. You need to be able to convert between them. The key is to remember the approximate conversion factors. One mile is longer than one kilometre, so when you convert from miles to kilometres, the number will get bigger. When you convert from kilometres to miles, the number will get smaller.

1 mile ≈ 1.6 kilometres

1 kilometre ≈ 5/8 of a mile (or ≈ 0.625 miles)

To convert miles to km: multiply by 1.6

To convert km to miles: divide by 1.6 (or multiply by 5/8)

Key term

Conversion Factor: A number used to change one set of units to another by multiplying or dividing.

Examiner insight

Examiners look for the correct application of the conversion factor. Clearly stating 'miles to km, so I multiply by 1.6' can help you structure your answer and gain marks even if you make a small arithmetic error.

Common pitfall

Multiplying when you should divide, or vice versa. To avoid this, remember that miles are longer, so there are fewer of them for a given distance compared to kilometres.

Fun fact

The 'mile' originated from the Roman 'mille passus', meaning 'a thousand paces'. Each pace was considered to be five Roman feet.

Worked example 12 marks

A signpost says the next town is 25 miles away. How far is this in kilometres?

  1. 1

    Identify the conversion needed: miles to kilometres.

  2. 2

    Recall the conversion rule: To convert miles to km, multiply by 1.6.

  3. 3

    Perform the calculation: 25 × 1.6.

  4. 4

    25 × 1 = 25. 25 × 0.6 = 15.

  5. 5

    25 + 15 = 40.

  6. 6

    The distance is 40 km.

Worked example 22 marks

A marathon is a 42.2 km race. Approximately how many miles is this? Use the conversion 1 km ≈ 5/8 mile.

  1. 1

    Identify the conversion needed: kilometres to miles.

  2. 2

    Recall the conversion rule: To convert km to miles, multiply by 5/8.

  3. 3

    Perform the calculation: 42.2 × 5/8.

  4. 4

    This is (42.2 × 5) ÷ 8.

  5. 5

    42.2 × 5 = 211.

  6. 6

    211 ÷ 8 = 26.375.

  7. 7

    The distance is approximately 26.4 miles.

Recap

  • To change miles to kilometres, multiply by 1.6.
  • To change kilometres to miles, divide by 1.6.
  • Using 1 km ≈ 5/8 mile is an alternative way to convert from km to miles.
  • There are more kilometres than miles for the same distance.
  • Always show which conversion factor you are using in your working.

Quick check

  1. Convert 50 miles into kilometres.1 mark
  2. Convert 160 kilometres into miles.1 mark

2. Area of Parallelograms and Trapezia

You can find the area of complex shapes by building on what you know about simpler ones. The formula for the area of a parallelogram is derived from a rectangle. If you cut a triangle from one end of a parallelogram and move it to the other, you form a rectangle with the same base and height. For a trapezium, you can derive the formula by thinking of it as half of a large parallelogram made by joining two identical trapezia together.

Area of a parallelogram = base × perpendicular height

Area of a trapezium = 1/2 × (sum of parallel sides) × perpendicular height

Area of a trapezium = 1/2(a+b)h

Key term

Perpendicular Height: The height of a shape measured at a right angle (90°) from the base to the highest point or opposite side.

Examiner insight

Examiners often include the slant height in diagrams to test whether you understand which measurement to use. Always look for the right-angle symbol to identify the perpendicular height.

Common pitfall

The most common mistake is using the length of the slanted side instead of the perpendicular height in calculations for both parallelograms and trapezia.

Worked example 12 marks

Calculate the area of a parallelogram with a base of 9 cm and a perpendicular height of 4 cm.

  1. 1

    State the formula for the area of a parallelogram: Area = base × perpendicular height.

  2. 2

    Substitute the given values into the formula: Area = 9 cm × 4 cm.

  3. 3

    Calculate the final area: Area = 36 cm².

  4. 4

    Remember to include the correct units (cm²) in your answer.

Worked example 23 marks

A trapezium has parallel sides of length 7 m and 11 m. The perpendicular height between the parallel sides is 6 m. Calculate the area of the trapezium.

  1. 1

    State the formula for the area of a trapezium: Area = 1/2(a+b)h.

  2. 2

    Identify the values: a=7, b=11 (the parallel sides), and h=6 (the perpendicular height).

  3. 3

    Substitute the values: Area = 1/2(7 + 11) × 6.

  4. 4

    Calculate the sum inside the brackets first: Area = 1/2(18) × 6.

  5. 5

    Complete the calculation: Area = 9 × 6 = 54 m².

Recap

  • The area of a parallelogram is its base multiplied by its perpendicular height.
  • The area of a trapezium is half the sum of its parallel sides, multiplied by the perpendicular height.
  • Always use the perpendicular height, not the length of a slanted side.
  • Area is measured in square units, like cm² or m².

Quick check

  1. A parallelogram has a base of 5m and a perpendicular height of 10m. What is its area?1 mark
  2. Find the area of a trapezium with parallel sides 4cm and 6cm, and a height of 5cm.2 marks

3. Volume of Triangular Prisms

A prism is a 3D shape that has a constant cross-section along its length. This means if you slice it at any point parallel to the ends, the sliced face is always the same shape and size. A triangular prism has a triangle as its cross-section. To find the volume of any prism, you calculate the area of its cross-section and multiply it by the length (or height) of the prism.

Volume of any prism = Area of cross-section × length

Area of a triangle = 1/2 × base × height

Volume of a triangular prism = (1/2 × b × h) × l

Key term

Cross-section: The 2D shape that is revealed when a 3D object is sliced straight through.

Examiner insight

Show your working in two distinct steps: calculating the cross-sectional area first, then multiplying by the length. This makes your method clear and can earn you partial marks if you make a mistake in the final calculation.

Common pitfall

Forgetting the '1/2' when calculating the area of the triangular cross-section is a very frequent error. Always double-check the triangle area calculation before finding the volume.

Worked example 13 marks

Calculate the volume of the triangular prism below. The triangular face has a base of 10 cm and a perpendicular height of 8 cm. The length of the prism is 15 cm.

  1. 1

    Step 1: Find the area of the triangular cross-section.

  2. 2

    Area of triangle = 1/2 × base × height = 1/2 × 10 cm × 8 cm = 40 cm².

  3. 3

    Step 2: Multiply the cross-sectional area by the length of the prism to find the volume.

  4. 4

    Volume = Area of cross-section × length = 40 cm² × 15 cm.

  5. 5

    Volume = 600 cm³.

  6. 6

    The volume is 600 cubic centimetres.

Worked example 22 marks

A chocolate bar is a triangular prism. Its volume is 105 cm³. The length of the bar is 14 cm. What is the area of the triangular end?

  1. 1

    State the formula for the volume of a prism: Volume = Area of cross-section × length.

  2. 2

    Rearrange the formula to find the area of the cross-section: Area = Volume ÷ length.

  3. 3

    Substitute the given values: Area = 105 cm³ ÷ 14 cm.

  4. 4

    Calculate the result: Area = 7.5 cm².

  5. 5

    The area of the triangular end is 7.5 cm².

Recap

  • A prism has a constant cross-section along its length.
  • The volume of any prism is found by multiplying the area of its cross-section by its length.
  • For a triangular prism, first find the area of the triangle (1/2 × base × height).
  • Then multiply the triangle's area by the prism's length.
  • Volume is measured in cubic units, like cm³ or m³.

Quick check

  1. The cross-section of a prism is a triangle with an area of 20 cm². If the prism is 5 cm long, what is its volume?1 mark
  2. A triangular prism has a volume of 100 cm³ and a length of 10 cm. What is the area of its cross-section?1 mark

4. Surface Area of 3D Shapes

The surface area of a 3D shape is the total area of all its outside surfaces. Imagine you 'unfold' the shape into a flat 2D pattern, called a net. The surface area is the total area of this net. The key is to be systematic: identify every face, calculate its individual area, and then add all the areas together.

Surface Area of a Cube = 6 × (side length)²

Surface Area of a Cuboid = 2(lw + lh + wh)

Surface area is the sum of the areas of all faces.

Key term

Net: A 2D pattern of a 3D shape that can be folded to make the object.

Examiner insight

A clear, systematic approach is highly valued. A good strategy is to list the faces you are calculating (e.g., 'Top/Bottom', 'Front/Back') and show the area for each before summing them. This demonstrates a logical method and helps prevent errors.

Common pitfall

Forgetting to include all the faces in the calculation. It's easy to miss the bottom or back faces that are not always clearly drawn in a 2D representation of a 3D shape.

Worked example 13 marks

A cuboid has a length of 7 cm, a width of 3 cm, and a height of 5 cm. Calculate its total surface area.

  1. 1

    Identify the three pairs of faces: front/back, top/bottom, left/right.

  2. 2

    Area of front and back faces = 2 × (length × height) = 2 × (7 cm × 5 cm) = 70 cm².

  3. 3

    Area of top and bottom faces = 2 × (length × width) = 2 × (7 cm × 3 cm) = 42 cm².

  4. 4

    Area of left and right faces = 2 × (width × height) = 2 × (3 cm × 5 cm) = 30 cm².

  5. 5

    Total Surface Area = 70 + 42 + 30 = 142 cm².

Worked example 24 marks

Find the surface area of a square-based pyramid. The square base has sides of 6 m, and the perpendicular height of each triangular face (the slant height) is 10 m.

  1. 1

    The shape has 5 faces: 1 square base and 4 identical triangular faces.

  2. 2

    Step 1: Calculate the area of the square base.

  3. 3

    Area of base = side × side = 6 m × 6 m = 36 m².

  4. 4

    Step 2: Calculate the area of one triangular face.

  5. 5

    Area of triangle = 1/2 × base × height = 1/2 × 6 m × 10 m = 30 m².

  6. 6

    Step 3: Calculate the total area of the four triangular faces.

  7. 7

    Area of 4 triangles = 4 × 30 m² = 120 m².

  8. 8

    Step 4: Add the area of the base and the area of the triangles.

  9. 9

    Total Surface Area = 36 m² + 120 m² = 156 m².

Recap

  • Surface area is the total area of all the faces of a 3D shape.
  • Break the problem down by identifying every face of the shape.
  • Calculate the area of each individual face.
  • Sum the areas of all the faces to find the total surface area.
  • Be systematic to ensure no faces are missed, especially hidden ones.

Quick check

  1. A cube has side lengths of 4 cm. What is its total surface area?2 marks
  2. A cereal box is a cuboid. How many faces does it have?1 mark

End-of-chapter exercise

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

  1. A car travels 120 miles. How far is this in kilometres? (Use 1 mile ≈ 1.6 km)2 marks
  2. A parallelogram has a base of 15 cm and a perpendicular height of 7 cm. Calculate its area.2 marks
  3. A field is shaped like a trapezium. Its parallel sides are 50 m and 80 m long. The perpendicular distance between these sides is 40 m. Calculate the area of the field.3 marks
  4. A block of cheese is a triangular prism. Its volume is 450 cm³. The area of its triangular end is 30 cm². What is the length of the block of cheese?2 marks
  5. Calculate the total surface area of a cuboid with length 10 cm, width 4 cm, and height 5 cm.3 marks
  6. A triangular prism has a cross-section that is a right-angled triangle with a base of 5 cm and a height of 12 cm. The length of the prism is 10 cm. Calculate the volume of the prism.3 marks
  7. A road race is 80 km long. Using the approximation 1 km ≈ 5/8 of a mile, calculate the length of the race in miles.2 marks
  8. The area of a trapezium is 108 cm². Its perpendicular height is 9 cm and one of its parallel sides is 10 cm. Find the length of the other parallel side.4 marks
  9. A tent is in the shape of a triangular prism. The triangular ends have a base of 1.6 m and a height of 1.2 m. The two slanted sides of the triangle are each 1.5 m long. The length of the tent is 2.5 m. Calculate the total surface area of the tent fabric, including the groundsheet (the base).5 marks
  10. A solid metal cube has a side length of 10 cm. A square-based pyramid has a base of 10 cm by 10 cm and a height such that its volume is exactly half the volume of the cube. What is the volume of the pyramid? (Note: you do not need the formula for the volume of a pyramid for this question).3 marks

Go deeper

Practise and revise with member-only material for this chapter.

Workbook

Exam-style practice questions with full worked solutions.

Unlock with a plan

Chapter at a Glance

A single-page summary of every key idea for fast revision.

Unlock with a plan

Free notes are just the start.

Unlock every Workbook and Chapter at a Glance, and generate your own worksheets and predicted papers.

Explore plans

Related chapters

Number (8Ni)Number (8Np)Number (8Nf)Algebra (8Ae)Algebra (8As)Geometry and Measure (8Gg)