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Geometry and Measure (6Gg)

Mathematics Stage 6 Chapter Notes

What this chapter covers

Geometry and Measure — Geometrical reasoning, shapes and measurements
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Geometry and Measure (6Gg) notes

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In this chapter

  1. 1Perimeter and Area of Rectangles
  2. 2Area and Perimeter of Compound Shapes
  3. 3Angles in a Triangle
  4. 4Plotting Coordinates
  5. 5Transformations: Reflection
  6. 63D Shapes and Nets

1. Perimeter and Area of Rectangles

Perimeter is the total distance around the outside of a 2D shape. Imagine walking along the edges of a field; the total distance you walk is the perimeter. It is measured in units of length like cm, m, or km. Area is the amount of surface a 2D shape covers. It's the space inside the shape, like the amount of grass in the field. Area is measured in square units, such as cm², m², or km².

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

Area of a rectangle = length × width

Key term

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

Common pitfall

Confusing perimeter and area. Students often mix up the formulas or use the wrong units (e.g., writing the perimeter in cm²).

Worked example 14 marks

A rectangular swimming pool is 12 metres long and 5 metres wide. Calculate its perimeter and area.

  1. 1
    1. Identify the length and width: length (l) = 12 m, width (w) = 5 m.
  2. 2
    1. Use the formula for perimeter: Perimeter = 2 × (l + w).
  3. 3
    1. Substitute the values: Perimeter = 2 × (12 + 5) = 2 × 17 = 34 m.
  4. 4
    1. Use the formula for area: Area = l × w.
  5. 5
    1. Substitute the values: Area = 12 × 5 = 60 m².
  6. 6

    Answer: The perimeter is 34 m and the area is 60 m².

Recap

  • Perimeter is the distance around a shape, found by adding all side lengths.
  • Area is the space inside a shape.
  • For a rectangle, Area = length × width.
  • For a rectangle, Perimeter = 2 × (length + width).
  • Remember to use the correct units: cm for perimeter, cm² for area.

Quick check

  1. A square has a side length of 6 cm. What is its area?1 mark
  2. A square has a side length of 6 cm. What is its perimeter?1 mark

2. Area and Perimeter of Compound Shapes

Compound shapes are made by joining two or more simple shapes together (like two rectangles). To find the area, the best method is to split the compound shape into its simple parts (e.g., rectangles), 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, as you may need to calculate the length of some unmarked sides first.

Key term

Compound Shape: A shape that is made up of two or more simpler geometric shapes.

Examiner insight

Examiners award marks for showing a clear method. Always draw lines on your diagram to show how you have split the compound shape, and write down the calculations for each part.

Fun fact

Architects and builders constantly calculate the area and perimeter of compound shapes when designing floor plans for houses or buildings.

Worked example 15 marks

Find the area and perimeter of the L-shaped figure shown. The lengths are in cm. The shape has a top edge of 10cm, a right edge of 7cm, a bottom edge of 4cm and a short right edge of 3cm.

  1. 1
    1. To find the area, split the shape into two rectangles. Let's make a vertical split. This creates a left rectangle (A) and a right rectangle (B).
  2. 2
    1. The missing side lengths must be found. The long left side is 7 cm. The short vertical side is 3 cm. The missing horizontal side is 10 cm - 4 cm = 6 cm.
  3. 3
    1. Calculate the area of rectangle A: It has dimensions 6 cm by 7 cm. Area A = 6 × 7 = 42 cm².
  4. 4
    1. Calculate the area of rectangle B: It has dimensions 4 cm by 3 cm. Area B = 4 × 3 = 12 cm².
  5. 5
    1. Total Area = Area A + Area B = 42 + 12 = 54 cm².
  6. 6
    1. To find the perimeter, add all the outside lengths: 10 + 7 + 4 + 3 + 6 + (7-3=4) = 34 cm. Let's recheck: 10 (top) + 7 (long right) + 4 (bottom) + 3 (inner vertical) + 6 (inner horizontal) is wrong. Add OUTSIDE lengths only: 10 + 7 + 4 + 3 + (10-4) + (7-3) = 10 + 7 + 4 + 3 + 6 + 4 = 34 cm.
  7. 7

    Answer: The area is 54 cm² and the perimeter is 34 cm.

Recap

  • To find the area of a compound shape, split it into simple rectangles.
  • Calculate the area of each rectangle and add them together.
  • To find the perimeter, add up the lengths of all the outside sides only.
  • You may need to calculate missing side lengths before you can find the perimeter.
  • Do not include the 'split' lines inside the shape when calculating the perimeter.

Quick check

  1. A T-shape is made from two rectangles. A 10cm by 2cm bar sits on top of a 2cm by 8cm post. What is the total area?2 marks

3. Angles in a Triangle

A fundamental rule in geometry is that the three interior angles in any triangle always add up to 180 degrees. This is true for all triangles, whether they are small or large, equilateral, isosceles, or scalene. Knowing this rule allows you to find a missing angle if you know the other two. For example, if two angles are 50° and 70°, the third must be 180° - 50° - 70° = 60°.

a + b + c = 180° (where a, b, and c are the three angles in a triangle)

Key term

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

Examiner insight

Always write down the rule 'Angles in a triangle sum to 180°' as part of your working. You can get a mark for showing you know the method, even if you make a calculation error.

Fun fact

This rule is only true on a flat, 2D surface. On a curved surface like a sphere, the angles of a triangle can add up to more than 180°! A triangle drawn from the North Pole down to the equator and back can have three 90° angles.

Worked example 12 marks

In triangle ABC, angle A is 45° and angle B is 80°. What is the size of angle C?

  1. 1
    1. State the rule: The sum of angles in a triangle is 180°.
  2. 2
    1. Add the known angles: 45° + 80° = 125°.
  3. 3
    1. Subtract the sum from 180° to find the missing angle: Angle C = 180° - 125°.
  4. 4
    1. Calculate the result: Angle C = 55°.

Worked example 23 marks

An isosceles triangle has one angle of 100°. What are the other two angles?

  1. 1
    1. In an isosceles triangle, two angles are equal. The 100° angle cannot be one of the equal pair, because 100°+100° is already more than 180°.
  2. 2
    1. Therefore, the 100° angle is the unique angle. The other two angles are the equal 'base' angles.
  3. 3
    1. Sum of angles = 180°. Amount remaining for the two equal angles = 180° - 100° = 80°.
  4. 4
    1. The two equal angles share this 80°. So, each angle is 80° ÷ 2 = 40°.
  5. 5
    1. Answer: The other two angles are 40° and 40°.

Recap

  • The three angles in any triangle always add up to 180°.
  • An equilateral triangle has 3 equal sides and 3 equal angles of 60°.
  • An isosceles triangle has 2 equal sides and 2 equal base angles.
  • A right-angled triangle has one angle that is exactly 90°.
  • If you know two angles, you can always find the third.

Quick check

  1. Two angles in a triangle are 30° and 70°. What is the third angle?1 mark

4. Plotting Coordinates

Coordinates are used to describe a precise position on a grid. The grid is called a Cartesian plane and has two axes: a horizontal x-axis and a vertical y-axis. The point where they cross is called the origin (0,0). Coordinates are always written as an ordered pair (x, y). The first number(x) tells you how far to go across (right for positive, left for negative), and the second number(y) tells you how far to go up or down (up for positive, down for negative). A good way to remember this is 'along the corridor, then up/down the stairs'.

Point = (x, y)

Key term

Quadrant: One of the four regions that the x-axis and y-axis divide the Cartesian plane into.

Examiner insight

Examiners look for accurately plotted points. Use a sharp pencil and take care to place your points exactly on the grid intersections.

Common pitfall

The most common mistake is mixing up the x and y coordinates. Students often plot the y-coordinate first, going up/down and then across. Always go across first.

Worked example 13 marks

The vertices of a quadrilateral are A(2, 3), B(-4, 3), C(-4, -1), and D(2, -1). Plot these points on a grid and name the shape.

  1. 1
    1. Draw x and y axes from at least -5 to 5.
  2. 2
    1. To plot A(2, 3): start at the origin, go 2 units right along the x-axis, then 3 units up.
  3. 3
    1. To plot B(-4, 3): start at the origin, go 4 units left, then 3 units up.
  4. 4
    1. To plot C(-4, -1): start at the origin, go 4 units left, then 1 unit down.
  5. 5
    1. To plot D(2, -1): start at the origin, go 2 units right, then 1 unit down.
  6. 6
    1. Join the points A to B, B to C, C to D, and D to A.
  7. 7
    1. Observe the shape. It has four right angles and opposite sides are equal length. It is a rectangle.

Recap

  • Coordinates are written in the form (x, y).
  • The x-axis is horizontal, the y-axis is vertical.
  • The origin is the point (0,0).
  • The first number (x) is the horizontal movement.
  • The second number (y) is the vertical movement.
  • Remember: 'along the corridor (x), then up the stairs (y)'.

Quick check

  1. What are the coordinates of a point that is 3 units left of the origin and 5 units up?1 mark

5. Transformations: Reflection

A reflection is a 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 congruent to the object, meaning it is the same size and shape, but its orientation is reversed.

Key term

Mirror Line: The fixed line over which a shape is flipped during a reflection.

Examiner insight

In an exam, you can use tracing paper. Trace the object and the mirror line, then flip the paper over the line to see exactly where the image should be drawn. This is a very reliable method.

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(4, 2), and R(1, 4). Reflect this triangle in the x-axis and state the coordinates of the image vertices P', Q', and R'.

  1. 1
    1. Identify the mirror line: the x-axis (the line where y=0).
  2. 2
    1. For each vertex, find its reflection. The x-coordinate will stay the same, and the y-coordinate will become its opposite.
  3. 3
    1. Point P is (1, 2). It is 2 units above the x-axis. The reflection P' will be 2 units below the x-axis. So, P' is (1, -2).
  4. 4
    1. Point Q is (4, 2). It is 2 units above the x-axis. The reflection Q' will be 2 units below the x-axis. So, Q' is (4, -2).
  5. 5
    1. Point R is (1, 4). It is 4 units above the x-axis. The reflection R' will be 4 units below the x-axis. So, R' is (1, -4).
  6. 6
    1. The coordinates of the image are P'(1, -2), Q'(4, -2), and R'(1, -4).

Recap

  • A reflection is a 'flip' over a mirror line.
  • The reflected image is the same size and shape as the original.
  • Each point on the image is the same distance from the mirror line as the original point.
  • When reflecting in the x-axis, the point (x, y) becomes (x, -y).
  • When reflecting in the y-axis, the point (x, y) becomes (-x, y).

Quick check

  1. What are the coordinates of the point (5, -2) after it is reflected in the y-axis?1 mark

6. 3D Shapes and Nets

Three-dimensional (3D) shapes have length, width, and height. They are made up of faces (the flat surfaces), edges (where two faces meet), and vertices (the corners where edges meet). A 'net' is a 2D pattern that you can cut out and fold to make a 3D shape. For example, the net of a cube is made of six squares joined together. Being able to visualize how a net folds up is a key skill.

Euler's Formula: Faces + Vertices - Edges = 2

Key term

Net: A two-dimensional pattern that can be folded to create a three-dimensional shape.

Common pitfall

When drawing a net, students sometimes draw the faces in an arrangement that would cause them to overlap or leave a gap when folded.

Fun fact

There are 11 different distinct nets that can be folded into a cube. Trying to find them all is a classic geometry puzzle!

Worked example 14 marks

A triangular prism is shown. How many faces, vertices, and edges does it have? Check your answer using Euler's formula.

  1. 1
    1. Count the faces: There are 2 triangular faces (the ends) and 3 rectangular faces (the sides). Total faces = 5.
  2. 2
    1. Count the vertices: There are 3 vertices on the front triangle and 3 on the back triangle. Total vertices = 6.
  3. 3
    1. Count the edges: There are 3 edges on the front triangle, 3 on the back, and 3 connecting them. Total edges = 9.
  4. 4
    1. Check with Euler's Formula (F + V - E = 2): 5 + 6 - 9 = 11 - 9 = 2.
  5. 5
    1. The formula holds true, so the counts are correct.
  6. 6

    Answer: 5 faces, 6 vertices, and 9 edges.

Recap

  • 3D shapes have faces (flat surfaces), edges (lines), and vertices (corners).
  • A net is the 2D layout of a 3D shape.
  • A cube has 6 faces, 8 vertices, and 12 edges.
  • A triangular prism has 5 faces, 6 vertices, and 9 edges.
  • Euler's formula (F + V - E = 2) works for any convex polyhedron.

Quick check

  1. How many faces does a standard cuboid have?1 mark
  2. A 3D shape has 8 faces and 12 vertices. How many edges does it have?2 marks

End-of-chapter exercise

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

  1. A rectangular garden is 15 metres long and has a perimeter of 46 metres. What is the area of the garden?3 marks
  2. An isosceles triangle has a perimeter of 25 cm. The shortest side is 7 cm. What are the lengths of the other two sides?2 marks
  3. Find the area of the compound shape formed by joining a 5cm by 8cm rectangle and a 6cm by 4cm rectangle along one of the 4cm sides. What is the perimeter of the new shape?5 marks
  4. A triangle has vertices at A(-2, 1), B(3, 1), and C(-1, 4). Plot the triangle. Then, reflect the triangle in the y-axis and write down the coordinates of the new vertices A', B', and C'.4 marks
  5. A triangle has angles of (x)°, (x + 20)°, and (2x - 40)°. By forming an equation, find the value of x and the size of all three angles.4 marks
  6. A square-based pyramid has 5 faces and 5 vertices. Use Euler's formula to determine the number of edges it has.2 marks
  7. The points A(-3, -2), B(3, -2), and C(3, 4) are three vertices of a rectangle ABCD. Find the coordinates of the vertex D and calculate the area of the rectangle.4 marks
  8. The net of a cube is made from six identical squares. The total area of the net is 96 cm². What is the length of one edge of the cube?3 marks
  9. A shape has vertices (1,1), (4,1), (4,3), (2,3), (2,5), (1,5). Plot the shape and find its perimeter.5 marks
  10. Triangle P is reflected in the line x=1 to give triangle Q. The vertices of triangle P are (2,2), (4,2) and (2,5). What are the coordinates of the vertices of triangle Q?3 marks

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