Cambridge Lower Secondary CheckpointStage 7

Geometry and Measure (7Gp)

Mathematics Stage 7 Chapter Notes

What this chapter covers

Geometry and Measure - Position and transformation
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In this chapter

  1. 1Understanding Basic Angle Rules
  2. 2Line and Rotational Symmetry
  3. 3Recognising Congruent Shapes
  4. 4Calculating Area of Triangles and Compound Shapes
  5. 5Volume and Surface Area of Cuboids
  6. 6Properties of Polygons and Circles

1. Understanding Basic Angle Rules

In geometry, angles have special relationships. Angles on a straight line always add up to 180°. This is because a straight line is like half a turn. Angles that meet at a point, forming a full circle, always add up to 360°. When two straight lines cross, they form 'X' shape. The angles opposite each other in the 'X' are called vertically opposite angles, and they are always equal.

Angles on a straight line add up to 180°.

Angles at a point add up to 360°.

Vertically opposite angles are equal.

Key term

Vertically Opposite Angles: Angles opposite each other when two straight lines intersect; they are always equal in size.

Examiner insight

Examiners award marks for stating the correct geometric reason (e.g., 'angles on a straight line') alongside your calculation.

Common pitfall

Confusing angles on a straight line with angles at a point, leading to using 360° instead of 180° or vice versa.

Worked example 14 marks

In the diagram, two straight lines intersect. Angle 'a' is 115°. Find the size of angles 'b' and 'c'. Give reasons for your answers.

  1. 1

    Step 1: Find angle b. Angles 'a' and 'b' are on a straight line. So, a + b = 180°.

  2. 2

    115° + b = 180°

  3. 3

    b = 180° - 115° = 65°

  4. 4

    Reason: Angles on a straight line add up to 180°.

  5. 5

    Step 2: Find angle c. Angles 'a' and 'c' are vertically opposite.

  6. 6

    Therefore, c = a.

  7. 7

    c = 115°

  8. 8

    Reason: Vertically opposite angles are equal.

Recap

  • A straight line has an angle of 180°.
  • A full circle or angles around a point sum to 360°.
  • When two lines cross, the angles opposite each other are equal.
  • Always give a reason for your angle calculations in an exam.

Quick check

  1. An angle on a straight line is 45°. What is the size of the other angle?1 mark
  2. Three angles around a point are 90°, 130°, and x. What is x?1 mark

2. Line and Rotational Symmetry

Symmetry describes how balanced a shape is. A shape has line symmetry if you can draw a mirror line through it and one side is a perfect reflection of the other. Some shapes have multiple lines of symmetry. A shape has rotational symmetry if it looks the same after being rotated less than a full 360° turn. The 'order of rotational symmetry' is the number of times the shape fits onto itself in one full turn.

Key term

Order of Rotational Symmetry: The number of times a shape fits onto itself during a full 360° rotation.

Common pitfall

Mistaking the number of sides for the order of rotational symmetry in non-regular polygons, for example, a rectangle has 4 sides but rotational symmetry of order 2.

Fun fact

Many logos, like the Volkswagen and Target symbols, use rotational symmetry to be memorable and balanced.

Worked example 12 marks

Consider a regular hexagon. State:a) The number of lines of symmetry.b) The order of rotational symmetry.

  1. 1

    a) A regular hexagon has 6 equal sides and 6 equal angles. You can draw a line of symmetry from each vertex to the opposite vertex (3 lines). You can also draw a line from the midpoint of each side to the midpoint of the opposite side (3 lines). Total lines of symmetry = 3 + 3 = 6.

  2. 2

    b) To find the order of rotational symmetry, imagine rotating the hexagon about its centre. It will fit perfectly onto its original outline 6 times in a full 360° turn. Therefore, the order of rotational symmetry is 6.

Recap

  • Line symmetry is also known as reflectional or mirror symmetry.
  • The order of rotational symmetry is how many times a shape looks the same in a 360° turn.
  • A shape with no rotational symmetry has an order of 1.
  • For regular polygons, the number of sides equals the number of lines of symmetry and the order of rotational symmetry.

Quick check

  1. How many lines of symmetry does a rectangle have?1 mark
  2. What is the order of rotational symmetry of an equilateral triangle?1 mark

3. Recognising Congruent Shapes

Two shapes are 'congruent' if they are exactly the same size and shape. Think of them as clones. If you can move one shape by translating (sliding), reflecting (flipping), or rotating (turning) it to fit exactly on top of the other, then the shapes are congruent. All their corresponding sides will be the same length, and all their corresponding angles will be the same size.

Key term

Congruent: Two shapes are congruent if they have the exact same size and shape, meaning all corresponding sides and angles are equal.

Examiner insight

When asked to explain why shapes are congruent, you don't need complex proofs. Simply stating which sides and angles match is often enough at this level.

Common pitfall

Confusing 'congruent' (same size and shape) with 'similar' (same shape, different size). A small square and a large square are similar, but not congruent.

Worked example 12 marks

Triangle ABC has sides AB = 5 cm, BC = 7 cm, and angle ABC = 90°. Triangle XYZ has sides XY = 5 cm, YZ = 7 cm, and angle XYZ = 90°. Are the two triangles congruent? Explain your answer.

  1. 1

    Step 1: Compare the given information for both triangles.

  2. 2

    Side AB corresponds to side XY, and both are 5 cm.

  3. 3

    Side BC corresponds to side YZ, and both are 7 cm.

  4. 4

    Angle ABC corresponds to angle XYZ, and both are 90°.

  5. 5

    Step 2: Conclude based on the comparison.

  6. 6

    Yes, the triangles are congruent.

  7. 7

    Explanation: They are identical in size and shape. We can match two sides and the angle between them, which are all equal.

Recap

  • Congruent shapes are identical in size and shape.
  • Corresponding sides and angles of congruent shapes are equal.
  • You can test for congruency by translation, reflection, or rotation.
  • Shapes with the same area are not necessarily congruent.

Quick check

  1. Are all squares with a side length of 4 cm congruent to each other?1 mark

4. Calculating Area of Triangles and Compound Shapes

The area of a shape is the amount of space it covers. For a triangle, the area is found by multiplying half of its base by its perpendicular height. The 'perpendicular height' is crucial – it's the height that makes a right angle with the base. For more complex 'compound shapes', the strategy is to split them into simpler shapes you know, like rectangles and triangles. Calculate the area of each part separately, then add them together.

Area of a triangle = ½ × base × perpendicular height

Area of a rectangle = length × width

Key term

Perpendicular Height: The height of a shape measured at a right angle (90°) to its base.

Examiner insight

For compound shape problems, drawing lines on the diagram to show how you've split it and labelling each part can get you method marks, even if your final calculation is wrong.

Common pitfall

Using a slanted side length instead of the perpendicular height when calculating a triangle's area.

Worked example 14 marks

Calculate the area of the compound shape shown, which is made from a rectangle and a triangle.

  1. 1

    Step 1: Split the shape into a rectangle (A) and a triangle (B).

  2. 2

    Step 2: Calculate the area of the rectangle (A). The dimensions are 10 cm by 6 cm.

  3. 3

    Area of A = length × width = 10 cm × 6 cm = 60 cm².

  4. 4

    Step 3: Calculate the area of the triangle (B). The base of the triangle is 10 cm. The total height of the shape is 9 cm, and the rectangle's height is 6 cm, so the triangle's height is 9 - 6 = 3 cm.

  5. 5

    Area of B = ½ × base × height = ½ × 10 cm × 3 cm = 15 cm².

  6. 6

    Step 4: Add the areas together to find the total area.

  7. 7

    Total Area = Area of A + Area of B = 60 cm² + 15 cm² = 75 cm².

Recap

  • The formula for the area of a triangle is ½ × base × height.
  • Always use the perpendicular height for triangle area calculations.
  • Break compound shapes into simpler shapes like rectangles and triangles.
  • Calculate the area of each simple shape first, then add them up.
  • Remember to use square units (like cm² or m²) for area.

Quick check

  1. A triangle has a base of 8 cm and a perpendicular height of 5 cm. What is its area?2 marks

5. Volume and Surface Area of Cuboids

For 3D shapes, we measure volume and surface area. Volume is the amount of space inside the shape, like how much water a box can hold. For a cuboid (a box shape), you find it by multiplying its length, width, and height. Surface area is the total area of all the faces of the 3D shape, like how much wrapping paper you'd need to cover the box. You calculate the area of each face and add them all together.

Volume of a cuboid = length × width × height

Surface Area of a cuboid = 2 × (length × width + length × height + width × height)

Key term

Surface Area: The total area of the outside surfaces of a three-dimensional object.

Common pitfall

Forgetting to calculate all six faces for surface area, often only calculating three and forgetting to double them.

Fun fact

Shipping companies care deeply about volume to fit as many packages as possible into a container, while manufacturers care about surface area to minimise packaging costs.

Worked example 14 marks

A cereal box is a cuboid with dimensions: height 30 cm, width 20 cm, and depth 8 cm. Calculate:a) its volume,b) its surface area.

  1. 1

    a) Calculate the volume:

  2. 2

    Volume = length × width × height (or depth x width x height)

  3. 3

    Volume = 8 cm × 20 cm × 30 cm

  4. 4

    Volume = 4800 cm³

  5. 5

    b) Calculate the surface area. A cuboid has 3 pairs of identical faces: front/back, top/bottom, left/right.

  6. 6

    Area of front/back = 2 × (20 cm × 30 cm) = 2 × 600 cm² = 1200 cm²

  7. 7

    Area of top/bottom = 2 × (20 cm × 8 cm) = 2 × 160 cm² = 320 cm²

  8. 8

    Area of left/right = 2 × (30 cm × 8 cm) = 2 × 240 cm² = 480 cm²

  9. 9

    Total Surface Area = 1200 + 320 + 480 = 2000 cm²

Recap

  • Volume is the space inside a 3D shape, measured in cubic units (cm³, m³).
  • Surface area is the total area of the faces, measured in square units (cm², m²).
  • For a cuboid, Volume = l × w × h.
  • A cuboid has 6 faces in 3 matching pairs.
  • A cube is a special cuboid where all sides are equal.

Quick check

  1. What is the volume of a cube with side length 2 cm?1 mark
  2. How many faces does a cuboid have?1 mark

6. Properties of Polygons and Circles

A polygon is any 2D shape with straight sides. We name them based on their number of sides: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), etc. A 'regular' polygon has all sides equal and all angles equal. A circle is a special shape. Key parts are the 'centre', the 'radius' (distance from the centre to the edge), the 'diameter' (distance across the circle through the centre), and the 'circumference' (the perimeter). The diameter is always twice the length of the radius.

Diameter = 2 × Radius

Key term

Regular Polygon: A polygon where all sides are equal in length and all interior angles are equal in size.

Examiner insight

Being able to correctly name polygons up to a decagon (10 sides) and use circle terminology accurately is a fundamental skill that underpins many geometry questions.

Common pitfall

Confusing radius and diameter. Remember the diameter is the 'long one' all the way across, so it's the bigger measurement.

Worked example 12 marks

a) What is the name of a polygon with 8 sides?b) A circle has a radius of 7 cm. What is its diameter?

  1. 1

    a) A polygon with 8 sides is called an octagon.

  2. 2

    b) The diameter is twice the radius.

  3. 3

    Diameter = 2 × Radius = 2 × 7 cm = 14 cm.

Recap

  • Polygons are closed 2D shapes with straight sides.
  • A regular polygon has equal sides and equal angles.
  • The radius of a circle is the distance from the centre to the edge.
  • The diameter goes all the way across the circle through the centre.
  • The diameter is always double the radius.

Quick check

  1. What is the name of a 5-sided polygon?1 mark
  2. If a circle's diameter is 20m, what is its radius?1 mark

End-of-chapter exercise

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

  1. Two straight lines, AB and CD, intersect at point E. If angle AEC is 55°, what are the sizes of angle CEB and angle BED? You must give a reason for each answer.4 marks
  2. A cuboid has a length of 12 cm, a width of 5 cm, and a height of 6 cm. Calculate its total surface area.3 marks
  3. What is the order of rotational symmetry and the number of lines of symmetry of a regular pentagon?2 marks
  4. Calculate the area of the shape below, which is formed by a rectangle and a triangle. The total height is 10m, the rectangle has a height of 7m, and the width is 8m.4 marks
  5. Triangle PQR is congruent to triangle LMN. In triangle PQR, PQ = 8cm, QR = 10cm, and angle PQR = 65°. What is the length of side LM and the size of angle LMN?2 marks
  6. A circle has a diameter of 9 cm. What is its radius?1 mark
  7. A large cube has a volume of 64 cm³. What is the length of one of its sides?2 marks
  8. Three angles meet at a point. Two of the angles are 110° and 140°. What is the size of the third angle?2 marks
  9. A shape is described as a quadrilateral with rotational symmetry of order 2 and two lines of symmetry. Name two possible shapes it could be.2 marks
  10. A triangular garden has a base of 15 metres and a perpendicular height of 8 metres. Calculate the area of the garden.2 marks

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