Cambridge Lower Secondary CheckpointStage 6

Geometry and Measure (6Gp)

Mathematics Stage 6 Chapter Notes

What this chapter covers

Geometry and Measure — Position and transformation
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Geometry and Measure (6Gp) notes

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In this chapter

  1. 1Perimeter of 2D Shapes
  2. 2Area of Rectangles and Squares
  3. 3Area and Perimeter of Compound Shapes
  4. 4Understanding Angles in a Triangle
  5. 5Working with Coordinates
  6. 6Transformations: Reflecting Shapes

1. Perimeter of 2D Shapes

Perimeter is the total distance around the outside of a two-dimensional (2D) shape. Imagine you are walking around the edge of a field; the total distance you walk is its perimeter. To find the perimeter of any shape with straight sides (a polygon), you simply add up the lengths of all its sides. It's important to make sure all lengths are in the same unit before you start adding.

Perimeter of any polygon = Sum of the lengths of all sides

Perimeter of a rectangle = 2 × (length + width)

Perimeter of a square = 4 × side length

Key term

Perimeter: The total distance around the boundary of a two-dimensional shape.

Examiner insight

Examiners expect you to show your working clearly. Even for simple shapes, write down the formula you are using and the numbers you are adding together.

Common pitfall

Confusing perimeter with area. Remember, perimeter is a length (measured in cm, m) while area is a space (measured in cm², m²).

Worked example 13 marks

A rectangular park is 45 metres long and 20 metres wide. A path runs around the edge of the park. If Sarah walks around the path once, how far has she walked?

  1. 1

    The shape is a rectangle, so we need to find its perimeter.

  2. 2

    The formula for the perimeter of a rectangle is P = 2 × (length + width).

  3. 3

    Substitute the given values: P = 2 × (45 m + 20 m).

  4. 4

    First, add the numbers in the brackets: 45 + 20 = 65 m.

  5. 5

    Now, multiply by 2: P = 2 × 65 = 130 m.

  6. 6

    So, Sarah has walked 130 metres.

Worked example 22 marks

A square has a perimeter of 36 cm. What is the length of one of its sides?

  1. 1

    A square has four equal sides.

  2. 2

    The formula for the perimeter of a square is P = 4 × side length.

  3. 3

    We are given the perimeter, P = 36 cm.

  4. 4

    So, 36 = 4 × side length.

  5. 5

    To find the side length, we need to divide the perimeter by 4.

  6. 6

    Side length = 36 cm ÷ 4 = 9 cm.

  7. 7

    The length of one side is 9 cm.

Recap

  • Perimeter is the total distance around the outside of a shape.
  • To find the perimeter of a polygon, add the lengths of all its sides.
  • The formula for the perimeter of a rectangle is 2 × (length + width).
  • The formula for the perimeter of a square is 4 × side length.
  • Always include the units (like cm, m, km) in your final answer.

Quick check

  1. Calculate the perimeter of a triangle with side lengths 5 cm, 7 cm, and 8 cm.1 mark
  2. What is the perimeter of a rectangle that is 10m long and 4m wide?1 mark

2. Area of Rectangles and Squares

Area is the measure of the amount of space inside a 2D shape. Think of it as the amount of paint needed to cover a surface or the number of tiles needed to cover a floor. Area is always measured in square units, such as square centimetres (cm²) or square metres (m²). For rectangles and squares, calculating the area is straightforward.

Area of a rectangle = length × width

Area of a square = side × side = side²

Key term

Area: The measure of the amount of space inside a two-dimensional shape, measured in square units.

Examiner insight

Examiners award marks for correctly stating the formula, showing the substitution of values, and giving the final answer with the correct units (e.g., cm²).

Common pitfall

Forgetting to write the units as squared units (e.g., writing 'cm' instead of 'cm²'). This can lose you a mark.

Worked example 12 marks

A smartphone screen measures 15 cm in length and 7 cm in width. Calculate the area of the screen.

  1. 1

    The screen is rectangular.

  2. 2

    The formula for the area of a rectangle is Area = length × width.

  3. 3

    Substitute the given values: Area = 15 cm × 7 cm.

  4. 4

    Calculate the product: 15 × 7 = 105.

  5. 5

    The area is 105 cm². Remember to use square units.

Worked example 22 marks

A square-shaped garden has a side length of 6 metres. What is the area of the garden?

  1. 1

    The garden is a square.

  2. 2

    The formula for the area of a square is Area = side².

  3. 3

    Substitute the given value: Area = 6 m × 6 m.

  4. 4

    Calculate the result: Area = 36 m².

  5. 5

    The area of the garden is 36 square metres.

Recap

  • Area measures the space inside a 2D shape.
  • Area is measured in square units (e.g., cm², m²).
  • The area of a rectangle is found by multiplying its length by its width.
  • The area of a square is the side length multiplied by itself.
  • Always state the correct square units in your answer.

Quick check

  1. A piece of paper is 30 cm long and 20 cm wide. What is its area?1 mark
  2. What is the area of a square with a side of 10 mm?1 mark

3. Area and Perimeter of Compound Shapes

Compound shapes (or composite shapes) are figures made by joining two or more simple shapes together, like rectangles. To find the area of a compound shape, you can split it into simpler shapes, calculate the area of each part, and then add them together. To find the perimeter, you must add up the lengths of all the outside edges. Be careful here: you might need to calculate the length of some 'missing' sides first.

Key term

Compound Shape: A 2D shape that is made up of two or more simpler shapes, such as rectangles or squares.

Common pitfall

When calculating the perimeter of a compound shape, a very common mistake is to include the 'internal' split line in the calculation. Only add the lengths of the outer boundary.

Fun fact

Architects and builders work with compound shapes all the time when designing floor plans for houses and buildings.

Worked example 15 marks

The L-shaped diagram shows a room. All corners are right angles. Find(a) the perimeter of the room, and(b) the area of the room.

  1. 1

    The diagram is an L-shape with outer dimensions 10m (top), 8m (right), and shorter sides of 4m and 3m.

  2. 2

    (a) To find the perimeter, we need the lengths of all outside edges. Let's find the two missing lengths.

  3. 3

    Missing horizontal length = 10 m - 4 m = 6 m.

  4. 4

    Missing vertical length = 8 m - 3 m = 5 m.

  5. 5

    Perimeter = 10 + 8 + 4 + 5 + 6 + 3 = 36 m.

  6. 6

    (b) To find the area, split the shape into two rectangles. Let's make a vertical split.

  7. 7

    Rectangle A (left): 6 m wide and 3 m high. Area A = 6 m × 3 m = 18 m².

  8. 8

    Rectangle B (right): 4 m wide and 8 m high. Area B = 4 m × 8 m = 32 m².

  9. 9

    Total Area = Area A + Area B = 18 m² + 32 m² = 50 m².

  10. 10

    Alternatively, make a horizontal split.

  11. 11

    Rectangle C (top): 10 m wide and 3 m high. Area C = 10 m × 3 m = 30 m².

  12. 12

    Rectangle D (bottom): 4 m wide and 5 m high. Area D = 4 m × 5 m = 20 m².

  13. 13

    Total Area = Area C + Area D = 30 m² + 20 m² = 50 m². The answer is the same.

Recap

  • Compound shapes are made from simpler shapes joined together.
  • To find the area, split the shape into rectangles, find the area of each, and add them up.
  • To find the perimeter, you must find the length of every outside edge and add them together.
  • Be careful to find any missing side lengths before calculating the perimeter.
  • Do not add the perimeters of the smaller shapes; this is a common mistake.

Quick check

  1. An L-shape has a total height of 10cm and total width of 8cm. A corner 'cut-out' is a 3cm by 4cm rectangle. What are the lengths of the two smaller internal sides?2 marks

4. Understanding Angles in a Triangle

Triangles are one of the most fundamental shapes in geometry. A key rule you must know is that the three interior angles in any triangle always add up to 180°. This rule allows you to find a missing angle if you know the other two. Triangles can be classified by their sides and angles: equilateral (3 equal sides, 3 equal 60° angles), isosceles (2 equal sides, 2 equal base angles), and scalene (no equal sides or angles).

Sum of angles in a triangle = 180°

Key term

Isosceles Triangle: A triangle that has two sides of equal length and, consequently, two angles of equal size.

Examiner insight

Marks are often awarded for providing reasons for your calculations, such as writing 'angles in a triangle sum to 180°' or 'base angles of an isosceles triangle are equal' next to your working.

Fun fact

The triangle is the strongest geometric shape. This is why it's used extensively in construction for structures like bridges and roof trusses to provide stability and support.

Worked example 12 marks

A triangle has two angles measuring 50° and 70°. What is the size of the third angle?

  1. 1

    The sum of angles in a triangle is 180°.

  2. 2

    Add the two known angles: 50° + 70° = 120°.

  3. 3

    Subtract this sum from 180° to find the third angle.

  4. 4

    Third angle = 180° - 120° = 60°.

Worked example 23 marks

An isosceles triangle has one angle of 40° at its peak (between the two equal sides). What are the sizes of the other two angles?

  1. 1

    In an isosceles triangle, the two base angles (opposite the equal sides) are equal.

  2. 2

    The sum of all angles is 180°. First, subtract the known angle: 180° - 40° = 140°.

  3. 3

    This 140° is the sum of the two equal base angles.

  4. 4

    To find the size of one base angle, divide by 2: 140° ÷ 2 = 70°.

  5. 5

    So, the other two angles are both 70°.

Recap

  • The three angles in any triangle always sum to 180°.
  • An equilateral triangle has three 60° angles.
  • An isosceles triangle has two equal sides and two equal angles.
  • A scalene triangle has no equal sides and no equal angles.
  • A right-angled triangle has one angle that is exactly 90°.

Quick check

  1. Can a triangle have two right angles? Explain why or why not.2 marks
  2. If two angles of a triangle are 35° and 55°, what is the third angle?1 mark

5. Working with Coordinates

Coordinates are a pair of numbers that describe the exact position of a point on a grid, called a Cartesian plane. The grid has a horizontal line called the x-axis and a vertical line called the y-axis. The point where they cross is called the origin (0,0). Coordinates are always written in a specific order in brackets: (x, y). A simple way to remember the order is 'along the corridor (x-axis), then up or down the stairs (y-axis)'.

Key term

Coordinates: An ordered pair of numbers (x, y) that specifies the position of a point on a Cartesian plane relative to the origin.

Common pitfall

The most common mistake is mixing up the x and y coordinates. Students often plot (3, 5) by going 5 along and 3 up, which is incorrect. Always deal with the x-coordinate first.

Fun fact

Coordinates are used in GPS and mapping apps like Google Maps to pinpoint any location on Earth with incredible accuracy using latitude and longitude, which work just like x and y coordinates.

Worked example 13 marks

Plot the points A(2, 1), B(6, 1), and C(6, 4) on a grid. Find the coordinates of point D such that ABCD is a rectangle.

  1. 1

    Draw x and y axes. Label them from 0 to at least 6.

  2. 2

    Plot point A: Go 2 along the x-axis and 1 up the y-axis.

  3. 3

    Plot point B: Go 6 along the x-axis and 1 up the y-axis.

  4. 4

    Plot point C: Go 6 along the x-axis and 4 up the y-axis.

  5. 5

    To make a rectangle, the fourth point D must be vertically above A and horizontally level with C.

  6. 6

    The x-coordinate of D must be the same as A (which is 2).

  7. 7

    The y-coordinate of D must be the same as C (which is 4).

  8. 8

    Therefore, the coordinates of point D are (2, 4).

Recap

  • Coordinates are written as an ordered pair (x, y).
  • The first number (x) tells you how far to go horizontally along the x-axis.
  • The second number (y) tells you how far to go vertically along the y-axis.
  • The origin is the point (0, 0) where the axes cross.
  • Remember the order: 'along the corridor, then up the stairs'.

Quick check

  1. What are the coordinates of the origin?1 mark
  2. Which point is further to the right on a grid, A(3, 5) or B(5, 3)?1 mark

6. Transformations: Reflecting Shapes

A transformation is a way of changing a shape's position or size. Reflection is a type of transformation that 'flips' a shape over a line, called the mirror line. The reflected shape, called the image, is a mirror image of the original shape, called the object. Every point on the image is the same distance from the mirror line as the corresponding point on the object, but on the opposite side. The image is the same size and shape as the object.

Key term

Mirror Line: The line across which a shape is reflected to create a mirror image; it acts like a mirror.

Examiner insight

Examiners look for accuracy when you draw the reflected shape. Use a pencil and ruler, and physically count the squares from each vertex to the mirror line to avoid errors.

Fun fact

The word 'Ambulance' is often written in reverse on the front of the vehicle so that drivers looking in their rear-view mirror can read it correctly.

Worked example 13 marks

A triangle has vertices at P(1, 2), Q(3, 2), and R(1, 4). Reflect this triangle in the y-axis and write down the coordinates of the new vertices P', Q', and R'.

  1. 1

    The y-axis is the vertical line where x=0. This is our mirror line.

  2. 2

    To reflect a point, count how many squares it is from the mirror line and count the same number of squares on the other side.

  3. 3

    Point P(1, 2) is 1 unit to the right of the y-axis. Its reflection, P', will be 1 unit to the left, at (-1, 2).

  4. 4

    Point Q(3, 2) is 3 units to the right of the y-axis. Its reflection, Q', will be 3 units to the left, at (-3, 2).

  5. 5

    Point R(1, 4) is 1 unit to the right of the y-axis. Its reflection, R', will be 1 unit to the left, at (-1, 4).

  6. 6

    The new coordinates are P'(-1, 2), Q'(-3, 2), and R'(-1, 4).

Worked example 24 marks

A square has vertices at A(2, 3), B(4, 3), C(4, 5) and D(2, 5). Reflect the square in the line y=2. What are the coordinates of the image?

  1. 1

    The mirror line is the horizontal line y=2.

  2. 2

    Point A(2, 3) is 1 unit above the line y=2. Its reflection A' will be 1 unit below it. A' is at (2, 1).

  3. 3

    Point B(4, 3) is 1 unit above the line y=2. Its reflection B' will be 1 unit below it. B' is at (4, 1).

  4. 4

    Point C(4, 5) is 3 units above the line y=2. Its reflection C' will be 3 units below it. C' is at (4, -1).

  5. 5

    Point D(2, 5) is 3 units above the line y=2. Its reflection D' will be 3 units below it. D' is at (2, -1).

  6. 6

    The new vertices are A'(2, 1), B'(4, 1), C'(4, -1), and D'(2, -1).

Recap

  • Reflection is a 'flip' transformation over a mirror line.
  • The reflected image is the same size and shape as the original object.
  • Every point on the image is the same perpendicular distance from the mirror line as the original point.
  • The y-axis is the line x=0.
  • The x-axis is the line y=0.
  • To reflect a point, count the squares to the mirror line and then count the same distance on the other side.

Quick check

  1. If the point (5, 4) is reflected in the x-axis, what are its new coordinates?1 mark
  2. If the point (2, 6) is reflected in the y-axis, what are its new coordinates?1 mark

End-of-chapter exercise

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

  1. A rectangle has a length of 15 cm and a width of 8 cm. Calculate its perimeter and area.3 marks
  2. A triangle has angles of 45° and 80°. What is the size of the third angle? What type of triangle is this (scalene, isosceles, or equilateral)?3 marks
  3. Plot the points P(-2, 1), Q(3, 1), and R(3, 4) on a coordinate grid. What are the coordinates of the point S that would make PQRS a rectangle?3 marks
  4. Calculate the area and perimeter of the L-shaped figure shown, where all corners are right angles. The outer dimensions are 12m by 9m, and the inner corner sides are 7m and 4m.5 marks
  5. A triangle has vertices at A(2, 5), B(4, 2), and C(1, 1). Reflect this triangle in the x-axis. State the coordinates of the new vertices A', B', and C'.3 marks
  6. A square photo frame has a perimeter of 80 cm. What is the area of the photo that can fit inside it?3 marks
  7. A rectangular garden measures 10 metres by 6 metres. A concrete path 1 metre wide is laid around the outside of the garden. Calculate the area of the concrete path.4 marks
  8. An isosceles triangle has a perimeter of 25 cm. The length of its longest side is 9 cm. What are the lengths of the other two sides?3 marks
  9. The vertices of a quadrilateral are at A(-3, -1), B(4, -1), C(4, 2), and D(-3, 2). a) What is the name of this shape? b) The shape is reflected in the y-axis. What are the coordinates of the vertex C'?3 marks
  10. An L-shaped room has a total area of 52 m². The shape is formed from two joined rectangles. The larger rectangle is 8m long and 5m wide. The smaller rectangle has a width of 3m. What is the length of the smaller rectangle?4 marks

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