Ch 2 · 4.4–4.5 Similarity and Symmetry · IGCSE Mathematics 0580
Topic 2 of 4 in Geometry — IGCSE Mathematics 0580 — 58 lessons.
Introduction
A model car, a photo enlargement, a large and a small bottle of the same design: these are similar shapes. If lengths double, areas do not double, they quadruple, and volumes go up eight times. That surprise is the heart of this lesson, which also covers symmetry in 2D and 3D. (Syllabus sections 4.4 and 4.5.)
Key ideas — Similar shapes and the scale factor
Two shapes are similar if one is an enlargement of the other: corresponding angles are equal and corresponding lengths are in the same ratio, the scale factor k:
🧑🏫 Teacher's explanation
Imagine you are using a photocopier to enlarge a photo. The photo doesn't change its shape; it just gets bigger. Every single line in that photo—the width, the height, the diagonal—gets multiplied by the exact same number. This number is what we call the scale factor, k. If you double the width, you must double the height, otherwise, the person in the photo would look stretched and weird!
📖 Definitions
🔍 Worked example — Easy
Two similar triangles, ABC and DEF, have corresponding sides AB = 5 cm and DE = 15 cm. Calculate the length of side EF if BC = 4 cm.
🔍 Worked example — Standard
Two similar cylinders have heights of 4 cm and 12 cm. Find the ratio of their volumes.
🔍 Worked example — Exam-style
A small sphere has a surface area of 10 cm^2. A larger, similar sphere has a surface area of 90 cm^2. Calculate the ratio of their volumes.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Showing triangles are similar
Two triangles are similar if they have two pairs of equal angles (the third pair then also matches). Give geometric reasons, e.g. "angle A is common; angle ADE = angle ABC (corresponding angles, DE parallel to BC)". Then match the vertices in order before writing ratios.
🧑🏫 Teacher's explanation
Imagine you are looking at a photograph of a triangle. If you zoom in on your phone screen, the triangle gets bigger, but its shape remains exactly the same. This is the essence of similarity. The angles don't change; only the side lengths do. In geometry, we say two shapes are 'similar' if they are identical in shape but different in size.
📖 Definitions
🔍 Worked example — Easy
Triangle ABC has angles 40^°, 60^°, and 80^°. Triangle DEF has angles 40^°, 60^°, and 80^°. Are they similar? If side AB = 5 cm and side DE = 10 cm, find the scale factor.
🔍 Worked example — Standard
In △ ABC, DE ∥ BC where D is on AB and E is on AC. If AD = 4 cm, AB = 12 cm, and DE = 3 cm, find the length of BC.
🔍 Worked example — Exam-style
Two similar triangles have areas of 25 cm^2 and 100 cm^2. Find the ratio of their corresponding side lengths.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Area and volume factors
If the length scale factor is k:
🧑🏫 Teacher's explanation
Imagine you have a small photo of a celebrity. If you decide to print a poster that is twice as wide and twice as tall as the photo, you aren't just using twice as much paper. Because you increased both the width and the height, the total area of the paper used actually quadruples. This is because area is a two-dimensional measurement (length × width).
📖 Definitions
🔍 Worked example — Easy
Two similar triangles, T_1 and T_2, have corresponding side lengths of 3 cm and 9 cm respectively. Calculate the ratio of their areas.
🔍 Worked example — Standard
Two similar spheres have surface areas in the ratio 25:49. Find the ratio of their volumes.
🔍 Worked example — Exam-style
A solid metal cone has a volume of 54 cm^3. A larger, similar cone has a volume of 1458 cm^3. Find the height of the larger cone if the height of the smaller cone is 4 cm.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Symmetry in 2D
🧑🏫 Teacher's explanation
Symmetry is all about balance and repetition. Imagine you are looking at a shape in a mirror; if the reflection looks exactly like the original, you have found a line of symmetry. It is like a folding instruction: if you can fold a piece of paper so the two halves match perfectly, that fold is a line of symmetry.
📖 Definitions
🔍 Worked example — Easy
State the number of lines of symmetry and the order of rotational symmetry for a regular pentagon.
🔍 Worked example — Standard
A shape is formed by two identical equilateral triangles joined at a common base to form a rhombus. Determine the number of lines of symmetry and the order of rotational symmetry.
🔍 Worked example — Exam-style
A shape has rotational symmetry of order 4. What is the minimum number of lines of symmetry this shape could have if it is a regular polygon?
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Symmetry in 3D
A plane of symmetry cuts a solid into two mirror-image halves; an axis of symmetry is a line the solid can be rotated about. A cuboid (not a cube) has 3 planes; a square-based pyramid has 4 planes and an axis of order 4; a cylinder and a cone have infinitely many planes through their axis.
🧑🏫 Teacher's explanation
Think about symmetry as 'balance' in a 3D object. If you could slice a shape perfectly in half so that one side is a mirror image of the other, that slice is a plane of symmetry. Imagine a knife cutting through a cake; if the two pieces are identical twins, you've found a plane of symmetry.
📖 Definitions
🔍 Worked example — Easy
Identify the number of planes of symmetry in a rectangular cuboid where the length, width, and height are all different.
🔍 Worked example — Standard
A square-based pyramid has a base with side length s and a vertex directly above the center of the base. Determine the number of planes of symmetry.
🔍 Worked example — Exam-style
A right circular cylinder has a height h and radius r. State the number of planes of symmetry and the order of rotational symmetry about its central axis.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
Triangles ABC and PQR are similar with AB = 4 cm and PQ = 10 cm. k = 2.5; if BC = 6 cm then QR = 15 cm. Area of PQR = 6.25 × area of ABC.
🔍 Worked example — Easy
Triangles ABC and PQR are similar. AB = 4 cm and PQ = 10 cm. If BC = 6 cm, find the length of QR and the ratio of their areas.
🔍 Worked example — Standard
Two similar bottles have heights of 12 cm and 18 cm. The smaller bottle has a volume of 400 cm³. Calculate the volume of the larger bottle.
🔍 Worked example — Exam-style
Two similar solids have surface areas of 45 cm² and 80 cm². If the smaller solid has a volume of 135 cm³, find the volume of the larger solid.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) k = 40/15 = tfrac83; height = 24 cm. (2) k = 2, area × 4 = 200 cm². (3) k = 3√(1000/64) = 10/4 = 2.5; height = 20 cm. (4) 8 lines, order 8.