Mathematics · Topic 1.3
Geometry
Geometry studies shapes, angles, and measurements. In this topic you will calculate perimeters, areas, and volumes, and explore the properties of 2D and 3D shapes.
What You'll Learn
- Classify angles as acute, right, obtuse, straight, and reflex
- Calculate the perimeter and area of rectangles, triangles, and parallelograms
- Find the circumference and area of circles using π
- Identify properties of common 2D shapes (polygons)
- Calculate the volume of rectangular prisms and cylinders
- Understand nets of 3D shapes
IB Assessment Focus
Criterion A: Select and apply correct formulas for perimeter, area, and volume.
Criterion B: Investigate patterns in angles and shapes; verify geometric rules.
Criterion C: Use correct units and present diagrams clearly.
Criterion D: Apply geometry to real-world contexts (e.g., packaging, floor plans).
Key Vocabulary
| Term | Definition |
|---|---|
| Perimeter | The total distance around the outside of a shape |
| Area | The amount of surface enclosed by a 2D shape (measured in square units) |
| Volume | The amount of space inside a 3D shape (measured in cubic units) |
| Polygon | A closed 2D shape with straight sides |
| Vertex | A corner point where two sides meet (plural: vertices) |
| Net | A 2D shape that can be folded to make a 3D shape |
| Radius | The distance from the centre of a circle to its edge |
| Diameter | The distance across a circle through its centre (= 2 × radius) |
Mathematics · Topic 1.3
Angles
An angle is formed where two lines or rays meet at a point. Angles are measured in degrees (°).
Types of Angles
| Type | Size | Description |
|---|---|---|
| Acute | Less than 90° | A small, "sharp" angle |
| Right | Exactly 90° | A quarter turn; marked with a small square |
| Obtuse | Between 90° and 180° | A "wide" angle, larger than a right angle |
| Straight | Exactly 180° | A half turn; a straight line |
| Reflex | Between 180° and 360° | More than a straight angle |
| Full turn | Exactly 360° | A complete rotation |
Angle Rules
- Angles on a straight line add up to 180°
- Angles at a point add up to 360°
- Angles in a triangle add up to 180°
- Angles in a quadrilateral add up to 360°
General Rule
Sum of interior angles of a polygon = (n − 2) × 180° (where n = number of sides)
Example: Find the missing angle in a triangle where two angles are 65° and 80°.
Angles in a triangle = 180°
Missing angle = 180 − 65 − 80 = 35°
Common Mistake: Students sometimes forget that angles in a triangle add up to 180°, not 360°. Remember: triangle = 180°, quadrilateral = 360°.
Mathematics · Topic 1.3
Perimeter & Area
Perimeter is the distance around a shape. Area is the amount of space inside it.
Key Formulas
| Shape | Perimeter | Area |
|---|---|---|
| Rectangle | P = 2(l + w) | A = l × w |
| Square | P = 4s | A = s² |
| Triangle | P = a + b + c | A = ½ × base × height |
| Parallelogram | P = 2(a + b) | A = base × height |
Perimeter Examples
Rectangle: Length = 12 cm, Width = 5 cm.
P = 2(12 + 5) = 2 × 17 = 34 cm
Square: Side = 9 cm.
P = 4 × 9 = 36 cm
Area Examples
Rectangle: Length = 8 m, Width = 3 m.
A = 8 × 3 = 24 m²
Triangle: Base = 10 cm, Height = 6 cm.
A = ½ × 10 × 6 = ½ × 60 = 30 cm²
Parallelogram: Base = 7 m, Height = 4 m.
A = 7 × 4 = 28 m²
Critical Rule: For a triangle, the height must be perpendicular (at 90°) to the base. The slanted side is NOT the height unless the triangle is right-angled.
Compound Shapes
For irregular shapes, split them into simpler shapes (rectangles, triangles), calculate each area separately, then add (or subtract) them.
Example: An L-shaped room can be split into two rectangles. Find each area and add them together for the total area.
Mathematics · Topic 1.3
Circles
A circle is a set of points all the same distance from the centre. The distance around a circle is called the circumference, and π (pi) is the special number that connects it to the diameter.
Circle Vocabulary
| Term | Definition |
|---|---|
| Radius (r) | Distance from the centre to any point on the circle |
| Diameter (d) | Distance across the circle through the centre; d = 2r |
| Circumference (C) | The perimeter of the circle |
| π (pi) | The ratio of circumference to diameter; π ≈ 3.14159… |
Circle Formulas
Circumference
C = 2πr = πd
Area
A = πr²
Examples
Circumference: A circle has radius 5 cm.
C = 2 × π × 5 = 10π ≈ 31.4 cm
Area: A circle has radius 5 cm.
A = π × 5² = 25π ≈ 78.5 cm²
Given the diameter: A circle has diameter 14 cm. Find the circumference.
r = 14 ÷ 2 = 7 cm
C = 2π(7) = 14π ≈ 44.0 cm
Common Mistake: If you are given the diameter, you must halve it to get the radius before using A = πr². Using the diameter as the radius will give an answer four times too large!
Mathematics · Topic 1.3
3D Shapes & Volume
Three-dimensional shapes have length, width, and height. Volume measures the space inside them.
Common 3D Shapes
| Shape | Faces | Edges | Vertices |
|---|---|---|---|
| Cube | 6 | 12 | 8 |
| Rectangular prism (cuboid) | 6 | 12 | 8 |
| Triangular prism | 5 | 9 | 6 |
| Cylinder | 3 (2 circles + 1 curved) | 2 | 0 |
| Cone | 2 (1 circle + 1 curved) | 1 | 1 |
| Sphere | 1 (curved) | 0 | 0 |
Volume Formulas
Rectangular Prism (Cuboid)
V = length × width × height
Cube
V = s³ (side length cubed)
Cylinder
V = πr²h
Examples
Rectangular prism: l = 8 cm, w = 4 cm, h = 3 cm.
V = 8 × 4 × 3 = 96 cm³
Cube: Side = 5 cm.
V = 5³ = 5 × 5 × 5 = 125 cm³
Cylinder: r = 3 cm, h = 10 cm.
V = π × 3² × 10 = π × 9 × 10 = 90π ≈ 282.7 cm³
Nets
A net is a 2D pattern that folds up to form a 3D shape. Understanding nets helps you visualise how 3D shapes are constructed.
- A cube net consists of 6 squares arranged so they fold into a cube
- A rectangular prism net has 6 rectangles (3 matching pairs)
- A cylinder net has 2 circles and 1 rectangle (whose width equals the circumference)
Critical Rule: Area is always in square units (cm², m²). Volume is always in cubic units (cm³, m³). Never mix units — convert everything to the same unit first.
Mathematics · Topic 1.3
Worked Examples
Step-by-step solutions showing the level of working expected in your assessments.
EXAMPLE 1Calculate the area of a triangle with base 10 cm and height 6 cm.
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Full Solution
A = ½ × base × height
A = ½ × 10 × 6
A = ½ × 60
A = 30 cm²
A = ½ × 10 × 6
A = ½ × 60
A = 30 cm²
EXAMPLE 2Find the missing angle in a triangle with angles of 42° and 73°.
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Full Solution
Angles in a triangle = 180°
Missing angle = 180 − 42 − 73 = 65°
Missing angle = 180 − 42 − 73 = 65°
EXAMPLE 3A circular garden has a diameter of 10 metres. Find (a) the circumference and (b) the area.
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Full Solution
(a) r = 10 ÷ 2 = 5 m. C = 2πr = 2 × π × 5 = 10π ≈ 31.4 m.
(b) A = πr² = π × 5² = 25π ≈ 78.5 m².
(b) A = πr² = π × 5² = 25π ≈ 78.5 m².
EXAMPLE 4A fish tank is 50 cm long, 30 cm wide, and 25 cm tall. What is its volume in litres?
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Full Solution
V = l × w × h = 50 × 30 × 25 = 37,500 cm³.
Since 1 litre = 1,000 cm³, the volume = 37,500 ÷ 1,000 = 37.5 litres.
Since 1 litre = 1,000 cm³, the volume = 37,500 ÷ 1,000 = 37.5 litres.
EXAMPLE 5A tin can has radius 4 cm and height 12 cm. Find the volume.
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Full Solution
V = πr²h = π × 4² × 12 = π × 16 × 12 = 192π ≈ 603.2 cm³.
EXAMPLE 6Describe the difference between a square and a rectangle.
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Full Solution
Both are quadrilaterals with four right angles (90°) and opposite sides that are parallel and equal. The key difference is that a square has all four sides equal, whereas a rectangle has two pairs of equal sides — the length and width can be different.
Mathematics · Topic 1.3
Practice Q&A
Attempt each question before revealing the model answer.
CALCULATEFind the perimeter and area of a rectangle with length 15 cm and width 8 cm.
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Model Answer
P = 2(15 + 8) = 2 × 23 = 46 cm.
A = 15 × 8 = 120 cm².
A = 15 × 8 = 120 cm².
CALCULATEA triangle has angles of 55° and 90°. Find the third angle.
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Model Answer
180 − 55 − 90 = 35°. The triangle is a right-angled triangle.
CALCULATEFind the area of a parallelogram with base 12 cm and perpendicular height 5 cm.
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Model Answer
A = base × height = 12 × 5 = 60 cm².
CALCULATEA circle has a radius of 7 cm. Find its circumference (use π ≈ 3.14).
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Model Answer
C = 2πr = 2 × 3.14 × 7 = 43.96 cm ≈ 44.0 cm.
CALCULATEFind the volume of a cube with side length 6 cm.
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Model Answer
V = s³ = 6³ = 6 × 6 × 6 = 216 cm³.
CLASSIFYClassify these angles: 35°, 90°, 150°, 200°.
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Model Answer
35° = acute (less than 90°). 90° = right angle. 150° = obtuse (between 90° and 180°). 200° = reflex (between 180° and 360°).
DESCRIBEHow many faces, edges, and vertices does a triangular prism have?
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Model Answer
A triangular prism has 5 faces (2 triangular ends + 3 rectangular sides), 9 edges, and 6 vertices.
CALCULATEFind the sum of interior angles in a hexagon (6 sides).
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Model Answer
Sum = (n − 2) × 180 = (6 − 2) × 180 = 4 × 180 = 720°.
Mathematics · Review
Flashcard Review
Tap each card to reveal the answer. Try to answer from memory first.
What is the formula for the area of a triangle?
A = ½ × base × height
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What is the formula for the circumference of a circle?
C = 2πr = πd
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What is the formula for the area of a circle?
A = πr²
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What do angles in a triangle add up to?
180°
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What do angles at a point add up to?
360°
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What is an obtuse angle?
An angle between 90° and 180°.
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Volume of a rectangular prism?
V = length × width × height
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Volume of a cylinder?
V = πr²h
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What is a net?
A 2D shape that can be folded to make a 3D shape.
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What is the relationship between diameter and radius?
Diameter = 2 × radius, or radius = diameter ÷ 2.
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Perimeter of a rectangle?
P = 2(length + width)
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What is a reflex angle?
An angle greater than 180° but less than 360°.
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What are the units for area?
Square units: cm², m², km², etc.
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What are the units for volume?
Cubic units: cm³, m³, etc. Also: 1 litre = 1,000 cm³.
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How many faces does a cube have?
6 faces (all squares).
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Mathematics · Practice Test
Practice Test — 20 Questions
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