Geometry & Measurement | Criticalyx Mathematics

🎯 Learning Objectives🎯 Objektif Pembelajaran

Calculate perimeter and area of 2D shapesKira perimeter dan luas bentuk 2D

Calculate surface area and volume of 3D shapesKira luas permukaan dan isipadu bentuk 3D

Apply circle theorems correctlyAplikasi teorem bulatan dengan betul

Find interior and exterior angles of polygonsCari sudut pedalaman dan sudut luaran poligon

Construct loci and interpret plans/elevationsBina lokus dan tafsir pelan/dongakan

📖 Key Concepts📖 Konsep Utama

2D Shapes (Form 1)Bentuk 2D (Tingkatan 1)

Perimeter and area of basic shapes.Perimeter dan luas bentuk asas.

Triangle: A = ½bhSegitiga: A = ½bt

Circle: A = πr², C = 2πrBulatan: A = πr², C = 2πr

Triangle: A = ½bh | Trapezium: A = ½(a+b)h | Circle: A = πr², C = 2πr
Sector: A = θ/360 × πr² | Arc length = θ/360 × 2πr

3D Shapes (Form 2)Bentuk 3D (Tingkatan 2)

Surface area and volume of solids.Luas permukaan dan isipadu pepejal.

Cylinder: V = πr²hSilinder: V = πr²t

Cone: V = ⅓πr²hKon: V = ⅓πr²t

Sphere: V = ⁴⁄₃πr³Sfera: V = ⁴⁄₃πr³

Pyramid: V = ⅓ × base × hPiramid: V = ⅓ × tapak × t

Cylinder: V = πr²h, CSA = 2πrh | Cone: V = ⅓πr²h
Sphere: V = ⁴⁄₃πr³, CSA = 4πr² | Pyramid: V = ⅓ × base area × h

Circles (Form 2)Bulatan (Tingkatan 2)

Circumference, arcs, sectors and circle theorems.Lilitan, arca, sektor dan teorem bulatan.

Sector & ArcSektor & Arca

Angle in semicircle = 90°Sudut dalam semibulatan = 90°

Angle at centre = 2 × angle at circumference
Angle in semicircle = 90°
Opposite angles of cyclic quadrilateral = 180°
Tangent ⊥ radius at point of contact

Polygons (Form 2)Poligon (Tingkatan 2)

Interior and exterior angle sums.Jumlah sudut pedalaman dan luaran.

Interior sum = (n − 2) × 180° | Exterior sum = 360°
Each exterior angle (regular) = 360° / n

✏️ Worked Examples✏️ Contoh Penyelesaian

Find the area of a sector with r = 7 cm, θ = 90°Cari luas sektor dengan r = 7 cm, θ = 90°

A = θ/360 × πr²

A = 90/360 × π(7²) = ¼ × 49π

= 12.25π = 38.5 cm²

38.5 cm²

Find the volume of a cone with r = 6 cm, h = 8 cmCari isipadu kon dengan r = 6 cm, h = 8 cm

V = ⅓πr²h = ⅓ × π × 36 × 8

V = 96π = 301.6 cm³

301.6 cm³

Find the interior angle of a regular pentagonCari sudut pedalaman pentagon biasa

Sum = (5 − 2) × 180° = 540°

Each = 540° / 5 = 108°

108°

Find arc length with r = 10 cm, θ = 72°Cari panjang arca dengan r = 10 cm, θ = 72°

Arc = θ/360 × 2πr

= 72/360 × 2π(10) = ⅕ × 20π = 4π

4π cm ≈ 12.57 cm

⚠️ Common Mistakes⚠️ Kesilapan Biasa

❌ Confusing area vs volume unitsMengelirukan unit luas vs isipadu

Area uses cm² (2D), Volume uses cm³ (3D). Never mix them up!Luas guna cm² (2D), Isipadu guna cm³ (3D). Jangan campurkan!

❌ Height vs slant height in cones/pyramidsTinggi vs tinggi miring dalam kon/piramid

Volume uses vertical height (h), NOT slant height (l). Use Pythagoras: l² = r² + h² for a cone.Isipadu guna tinggi menegak (h), BUKAN tinggi miring (l). Guna Pythagoras: l² = r² + h² untuk kon.

❌ Missing circle theorem conditionsTerlepas syarat teorem bulatan

Cyclic quadrilateral: all 4 vertices must be ON the circle. Tangent theorem: the angle is 90° only at the point of contact.Sisi empat kitaran: semua 4 bucu mesti PADA bulatan. Teorem tangen: sudut ialah 90° hanya pada titik sentuhan.

📝 SPM Practice📝 Amalan SPM

Paper 1 — Multiple ChoiceKertas 1 — Pilihan Berganda

Area of a circle with r = 14 cm (π = 22/7)Luas bulatan r = 14 cm (π = 22/7)

616 cm²

88 cm²

308 cm²

154 cm²

Volume of a cylinder: r = 3, h = 10Isipadu silinder: r = 3, h = 10

30π

90π

300π

9π

Interior angle of a regular hexagonSudut pedalaman heksagon biasa

108°

120°

135°

720°

Angle in a semicircle equalsSudut dalam semibulatan sama dengan

180°

90°

60°

360°

Arc length: r = 10, θ = 72°Panjang arca: r = 10, θ = 72°

4π

72π

20π

2π

Paper 2 — Structured QuestionsKertas 2 — Soalan Berstruktur

5 marks5 markah

A cone has base radius 5 cm and slant height 13 cm.
(a) Find its vertical height. [3 marks]
(b) Find its volume. [2 marks]Kon mempunyai jejari tapak 5 cm dan tinggi miring 13 cm.
(a) Cari tinggi menegaknya. [3 markah]
(b) Cari isipadunya. [2 markah]

Solution (a)Penyelesaian (a)

l² = r² + h² (Pythagoras)

13² = 5² + h² → 169 = 25 + h²

h² = 144 → h = 12 cm

h = 12 cm [3 marks3 markah]

Solution (b)Penyelesaian (b)

V = ⅓πr²h = ⅓ × π × 25 × 12

V = 100π = 314.2 cm³

314.2 cm³ [2 marks2 markah]

5 marks5 markah

In a circle with centre O, angle AOB = 120° and radius = 14 cm.
(a) Find the area of sector AOB. [3 marks]
(b) Find the length of arc AB. [2 marks]Dalam bulatan berpusat O, sudut AOB = 120° dan jejari = 14 cm.
(a) Cari luas sektor AOB. [3 markah]
(b) Cari panjang arca AB. [2 markah]

Solution (a)Penyelesaian (a)

A = θ/360 × πr² = 120/360 × π(196)

= ⅓ × 196π = 65⅓π

≈ 205.3 cm² [3 marks3 markah]

Solution (b)Penyelesaian (b)

Arc = θ/360 × 2πr = 120/360 × 2π(14)

= ⅓ × 28π = 28π/3

≈ 29.3 cm [2 marks2 markah]

📋 Formula Summary📋 Ringkasan Formula

2D Shapes:Bentuk 2D:
Triangle: A = ½bh | Circle: A = πr², C = 2πr
Sector: A = θ/360 × πr² | Arc = θ/360 × 2πr
Trapezium: A = ½(a+b)h

3D Shapes:Bentuk 3D:
Cylinder: V = πr²h, CSA = 2πrh
Cone: V = ⅓πr²h | Sphere: V = ⁴⁄₃πr³, CSA = 4πr²
Pyramid: V = ⅓ × base area × h

Polygons:Poligon:
Interior sum = (n − 2) × 180° | Exterior sum = 360°

Circle Theorems:Teorem Bulatan:
Centre = 2 × circumference | Semicircle = 90°
Cyclic quad opposite = 180° | Tangent ⊥ radius