Cambridge Lower Secondary CheckpointStage 7

Geometry and Measure (7Gg)

Mathematics Stage 7 Chapter Notes

What this chapter covers

Geometry and Measure - Geometrical reasoning, shapes and measurements
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In this chapter

  1. 1Fundamental Angle Rules
  2. 2Angles and Parallel Lines
  3. 3Angles in Polygons
  4. 4Circles: Area and Circumference
  5. 5Volume and Surface Area of Prisms
  6. 6Transformations: Reflection and Rotation

1. Fundamental Angle Rules

Understanding angles is foundational to geometry. There are three key rules you must know. First, angles on a straight line always add up to 180°. Second, angles that meet at a point, forming a full circle, always add up to 360°. Finally, when two straight lines intersect, they form two pairs of 'vertically opposite' angles. These opposite angles are always equal to each other.

Angles on a straight line = 180°

Angles around a point = 360°

Vertically opposite angles are equal

Key term

Vertically opposite angles: The angles that are directly opposite each other when two straight lines intersect.

Examiner insight

Examiners award marks for stating the geometric reason for your calculation, for example, writing '(angles on a straight line)' next to your working.

Common pitfall

Assuming angles are 90° just because they look like right angles. Only assume a 90° angle if it is marked with a square symbol.

Worked example 13 marks

In the diagram, three straight lines intersect at a single point. Find the values of angles x, y, and z.

  1. 1

    Angle y is vertically opposite the 40° angle. Therefore, y = 40°.

  2. 2

    Angles 95°, x, and y lie on a straight line. So, 95° + x + 40° = 180°.

  3. 3

    To find x, rearrange the equation: x = 180° - 95° - 40°.

  4. 4

    x = 45°.

  5. 5

    Angle z is vertically opposite angle x. Therefore, z = 45°.

Recap

  • Angles on a straight line sum to 180°.
  • Angles around a point sum to 360°.
  • Vertically opposite angles are equal.
  • Always state the angle rule you are using to justify your answer.

Quick check

  1. A straight line is split into two angles. One angle is 110°. What is the other angle?1 mark
  2. Two lines cross. One of the angles formed is 65°. What is the vertically opposite angle?1 mark

2. Angles and Parallel Lines

When a straight line, called a transversal, cuts across two parallel lines, it creates special pairs of angles. 'Alternate' angles are on opposite sides of the transversal and between the parallel lines, forming a 'Z' shape; they are always equal. 'Corresponding' angles are in the same position at each intersection, forming an 'F' shape; they are also equal. 'Co-interior' (or allied) angles are on the same side of the transversal and between the parallel lines, forming a 'C' or 'U' shape; they add up to 180°.

Alternate angles are equal ('Z' angles)

Corresponding angles are equal ('F' angles)

Co-interior angles add up to 180° ('C' angles)

Key term

Transversal: A line that intersects two or more other lines, typically parallel lines in exam questions.

Examiner insight

Marks are consistently lost for not providing a reason, or for giving a vague reason like 'Z-angle'. Use the correct terminology: 'alternate angles'.

Common pitfall

Confusing alternate and corresponding angles, or applying the rules when the lines are not marked as parallel.

Worked example 14 marks

In the diagram, line AB is parallel to line CD. Find the values of angles p and q, giving reasons for your answers.

  1. 1

    Angle p and the 125° angle are co-interior angles. Co-interior angles add up to 180°.

  2. 2

    p = 180° - 125° = 55°. Reason: Co-interior angles.

  3. 3

    Angle q and the 125° angle are alternate angles. Alternate angles are equal.

  4. 4

    q = 125°. Reason: Alternate angles. (Alternatively, p and q are angles on a straight line, so q = 180° - 55° = 125°).

Recap

  • Look for F-shapes for corresponding angles (equal).
  • Look for Z-shapes for alternate angles (equal).
  • Look for C-shapes for co-interior angles (add to 180°).
  • These rules only apply if the lines are parallel.
  • Always state the full name of the angle rule in your reasoning.

Quick check

  1. A transversal cuts two parallel lines. One corresponding angle is 70°. What is the other?1 mark

3. Angles in Polygons

A polygon is a 2D shape with straight sides. The sum of the interior angles depends on the number of sides(n) and can be found by splitting the polygon into triangles from one vertex. The sum of the exterior angles of any convex polygon is always 360°. A 'regular' polygon has all sides of equal length and all interior angles of equal size.

Sum of interior angles = (n - 2) × 180°

Sum of exterior angles = 360°

Interior angle + Exterior angle = 180°

Key term

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

Fun fact

Bees build their honeycombs using hexagons because it is the most efficient shape to tile a surface, using the least amount of wax to store the most honey.

Worked example 13 marks

The interior angle of a regular polygon is 144°. How many sides does the polygon have?

  1. 1

    First, find the exterior angle. Interior and exterior angles add to 180°.

  2. 2

    Exterior angle = 180° - 144° = 36°.

  3. 3

    The sum of all exterior angles is 360°. For a regular polygon, all exterior angles are equal.

  4. 4

    Number of sides(n) = Sum of exterior angles / Size of one exterior angle.

  5. 5

    n = 360° / 36° = 10.

  6. 6

    The polygon has 10 sides (it is a decagon).

Worked example 22 marks

Calculate the sum of the interior angles of a nonagon (9-sided polygon).

  1. 1

    Use the formula: Sum of interior angles = (n - 2) × 180°.

  2. 2

    Here, n = 9.

  3. 3

    Sum = (9 - 2) × 180° = 7 × 180°.

  4. 4

    Sum = 1260°.

Recap

  • The sum of interior angles is (n - 2) × 180°.
  • The sum of exterior angles is always 360°.
  • For regular polygons, all angles and sides are equal.
  • Finding the exterior angle first is often the easiest way to solve problems about regular polygons.

Quick check

  1. What is the sum of the interior angles of a pentagon (5 sides)?2 marks
  2. What is the size of one exterior angle of a regular hexagon (6 sides)?1 mark

4. Circles: Area and Circumference

A circle is defined by its centre and its radius (r), which is the distance from the centre to the edge. The diameter(d) is the distance across the circle through the centre (d = 2r). The circumference is the perimeter of the circle. The area is the space inside the circle. Both are calculated using pi (π), a special constant approximately equal to 3.142.

C = πd

C = 2πr

A = πr²

Key term

Pi (π): The constant ratio of a circle's circumference to its diameter, approximately 3.14159.

Examiner insight

Examiners will test if you can work backwards, for example, by giving you the area and asking for the radius or diameter. Show each step of your algebraic manipulation.

Common pitfall

Using the diameter instead of the radius in the area formula (A = πr²). Always halve the diameter first if needed.

Worked example 14 marks

A circle has a radius of 5 cm.a) Calculate its circumference.b) Calculate its area. Give both answers in terms of π.

  1. 1

    a) The formula for circumference is C = 2πr.

  2. 2

    Substitute r = 5 cm: C = 2 × π × 5.

  3. 3

    C = 10π cm.

  4. 4

    b) The formula for area is A = πr².

  5. 5

    Substitute r = 5 cm: A = π × (5)².

  6. 6

    A = 25π cm².

Worked example 23 marks

The area of a circle is 100π cm². What is its diameter?

  1. 1

    The formula for area is A = πr².

  2. 2

    We are given A = 100π. So, 100π = πr².

  3. 3

    Divide both sides by π: 100 = r².

  4. 4

    Find the square root: r = √100 = 10 cm.

  5. 5

    The question asks for the diameter. The diameter is twice the radius: d = 2r.

  6. 6

    d = 2 × 10 = 20 cm.

Recap

  • Circumference is the distance around the circle (C = πd).
  • Area is the space inside the circle (A = πr²).
  • The radius is half the diameter.
  • Always check if the question gives you the radius or the diameter.
  • Pay attention to whether the answer should be in terms of π or a decimal.

Quick check

  1. A circle has a diameter of 20 cm. What is its radius?1 mark
  2. A circle has a radius of 3 cm. What is its area in terms of π?1 mark

5. Volume and Surface Area of Prisms

A prism is a 3D shape that has a constant cross-section along its length. Think of a loaf of bread – every slice has the same shape. The volume of any prism is found by multiplying the area of its cross-section by its length (or height). The surface area is the total area of all the faces of the 3D shape added together. For a cuboid, which is a rectangular prism, the cross-section is a rectangle.

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

Volume of a cuboid = length × width × height

Surface area of a cuboid = 2(lw + lh + wh)

Key term

Cross-section: The 2D shape that is revealed when a 3D object is sliced through, which is constant along the length of a prism.

Common pitfall

Forgetting some of the faces when calculating surface area, especially the 'hidden' back, bottom, or side faces in a diagram.

Worked example 14 marks

A cuboid has dimensions of length 10 cm, width 4 cm, and height 5 cm. Calculate:a) its volume,b) its total surface area.

  1. 1

    a) Volume = length × width × height.

  2. 2

    Volume = 10 cm × 4 cm × 5 cm = 200 cm³.

  3. 3

    b) Surface area is the sum of the areas of the 6 faces. There are three pairs of identical faces: front/back, top/bottom, left/right.

  4. 4

    Area of front/back = 2 × (10 × 5) = 100 cm².

  5. 5

    Area of top/bottom = 2 × (10 × 4) = 80 cm².

  6. 6

    Area of left/right = 2 × (4 × 5) = 40 cm².

  7. 7

    Total Surface Area = 100 + 80 + 40 = 220 cm².

Worked example 23 marks

A triangular prism has a length of 12 cm. Its cross-section is a right-angled triangle with base 6 cm and height 8 cm. Find the volume of the prism.

  1. 1

    First, find the area of the cross-section, which is a triangle.

  2. 2

    Area of triangle = ½ × base × height = ½ × 6 × 8 = 24 cm².

  3. 3

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

  4. 4

    Volume = 24 cm² × 12 cm = 288 cm³.

Recap

  • Volume of any prism is the area of its cross-section multiplied by its length.
  • Surface area is the total area of all the outside faces.
  • Remember that volume is measured in cubic units (e.g., cm³) and area is in square units (e.g., cm²).
  • For surface area of a cuboid, find the area of the three different faces, add them up, and then double the result.

Quick check

  1. A cube has side length 3 cm. What is its volume?1 mark
  2. The cross-section of a prism has an area of 10 cm². Its length is 7 cm. What is its volume?1 mark

6. Transformations: Reflection and Rotation

Transformations change the position or size of a shape. A reflection flips a shape across a mirror line. Every point on the image is the same distance from the mirror line as the corresponding point on the original object. A rotation turns a shape around a fixed point, called the centre of rotation. To describe a rotation fully, you must state the centre of rotation, the angle of rotation (e.g., 90°, 180°), and the direction (clockwise or anticlockwise).

Key term

Invariant point: A point that does not move after a transformation has been applied.

Examiner insight

A common mistake is giving an incomplete description. For three marks on a rotation question, you must state all three elements: angle, direction, and centre.

Fun fact

The symmetry in logos, art, and nature, like the rotational symmetry of a starfish or the reflectional symmetry of a butterfly, are all examples of geometric transformations.

Worked example 12 marks

Triangle P has vertices at (1, 2), (1, 4), and (3, 2). Reflect Triangle P in the y-axis to get Triangle Q. State the coordinates of the vertices of Triangle Q.

  1. 1

    The y-axis is the line x=0. A reflection in the y-axis changes the sign of the x-coordinate of each point, while the y-coordinate stays the same. The rule is (x,y) -> (-x, y).

  2. 2

    Vertex (1, 2) becomes (-1, 2).

  3. 3

    Vertex (1, 4) becomes (-1, 4).

  4. 4

    Vertex (3, 2) becomes (-3, 2).

  5. 5

    The coordinates of the vertices of Triangle Q are (-1, 2), (-1, 4), and (-3, 2).

Worked example 23 marks

Describe fully the single transformation that maps Shape A onto Shape B.

  1. 1

    By observation, Shape B is a rotation of Shape A. Tracing paper can help confirm this.

  2. 2

    To find the centre of rotation, join corresponding points (e.g., top-right corner of A to top-left corner of B) and construct their perpendicular bisectors. The bisectors will intersect at the centre of rotation. In this case, the centre is the origin (0,0).

  3. 3

    To find the angle, pick a point on A, its image on B, and the centre. For example, point (2,1) on A moves to (-1,2) on B. This is a 90° turn.

  4. 4

    The direction is from quadrant 1 to quadrant 2, which is an anticlockwise direction.

  5. 5

    The full description is: Rotation, 90° anticlockwise, about the centre (0,0).

Recap

  • A reflection flips a shape across a mirror line.
  • A rotation turns a shape around a centre of rotation.
  • To fully describe a reflection, you only need the equation of the mirror line.
  • To fully describe a rotation, you need the centre, angle, and direction.
  • Using tracing paper is a very effective method for rotation problems in an exam.

Quick check

  1. What are the three pieces of information needed to fully describe a rotation?1 mark
  2. The point (4, -5) is reflected in the x-axis. What are the coordinates of its image?1 mark

End-of-chapter exercise

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

  1. In a diagram, two parallel lines are intersected by a transversal. One of the alternate interior angles is given as (3x + 10)° and the other is (5x - 50)°. Find the value of x and the size of the angles.4 marks
  2. A garden is shaped like a rectangle measuring 12m by 8m. A circular pond with a diameter of 4m is placed in the middle of the garden. Calculate the area of the garden that is covered by grass, to one decimal place.4 marks
  3. The sum of the interior angles of a polygon is 1800°. How many sides does the polygon have?2 marks
  4. A cylindrical tin of soup has a radius of 3.5 cm and a height of 11 cm. Calculate the volume of the tin. (Use π ≈ 22/7 or 3.142)3 marks
  5. Triangle A has vertices at P(2, 2), Q(5, 2), and R(2, 6). Describe fully the single transformation that maps triangle A onto triangle B with vertices at P'(-2, -2), Q'(-5, -2), and R'(-2, -6).3 marks
  6. A closed cardboard box is a cuboid measuring 30 cm by 20 cm by 10 cm. Calculate the total surface area of cardboard needed to make the box.3 marks
  7. An exterior angle of a regular polygon is 24°. How many sides does it have?2 marks
  8. Shape F is reflected in the line y = x to give shape G. Shape G is then rotated 180° about the origin to give shape H. If a point on F has coordinates (3, -1), what are the coordinates of the corresponding point on shape H?4 marks
  9. Find the size of the angle marked 'x' in a pentagon where the other four interior angles are 110°, 95°, 125°, and 80°.3 marks
  10. A water trough is a prism of length 3m. The cross-section is a trapezium with parallel sides of 60cm and 40cm, and a perpendicular height of 30cm. Calculate the volume of the trough in cubic metres.5 marks

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