Topic A: Cross Sections & 3D Figures
Lessons 1–5 · Identifying and analyzing cross sections.
01–03 — Lessons 1–3
Cross Sections of 3D Figures
- Cross section: the 2D shape formed when a plane intersects a 3D solid.
- Horizontal cross section of a cylinder → circle.
- Vertical cross section of a cylinder → rectangle.
- Diagonal cross section of a cube → rectangle or hexagon.
- Cross section of a cone → circle (horizontal) or triangle (vertical through apex).
- Cross section of a sphere → always a circle.
A square pyramid cut horizontally → smaller square. Cut vertically through apex → triangle.
Practice — Lessons 1–3
Q1 · Multiple Choice
A vertical cross section through the apex of a cone produces:
Q2 · Multiple Choice
A horizontal cross section of a cylinder is:
04–05 — Lessons 4–5
Nets & Surface Area Introduction
- Net: an unfolded 2D representation of a 3D solid.
- Each face of the 3D solid appears as a polygon in the net.
- Surface area = total area of all faces (add up the net's pieces).
- Cube net: 6 equal squares.
- Rectangular prism net: 3 pairs of rectangles.
A cube with side 4: SA = 6 × (4×4) = 96 sq units.
Practice — Lessons 4–5
Q3 · Multiple Choice
The net of a triangular prism consists of:
Q4 · Short Answer
A cube has side length 5. What is its surface area?
Topic B: Volume
Lessons 6–11 · Volume formulas for all standard 3D shapes.
06–08 — Lessons 6–8
Volume of Prisms & Cylinders
- Prism: V = Base Area × height = Bh.
- Rectangular prism (box): V = l × w × h.
- Triangular prism: V = (1/2)bh × length.
- Cylinder: V = πr²h.
- A prism has two congruent parallel bases. The lateral faces are rectangles.
Cylinder r=3, h=10: V = π(9)(10) = 90π ≈ 283 cu units.
Practice — Lessons 6–8
Q5 · Short Answer
Volume of a rectangular box: l=4, w=6, h=3.
Q6 · Multiple Choice
A cylinder has r = 5 and h = 8. Its volume is:
09–11 — Lessons 9–11
Volume of Pyramids, Cones & Spheres
- Pyramid: V = (1/3)Bh. (One-third of the prism with same base and height.)
- Cone: V = (1/3)πr²h. (One-third of cylinder.)
- Sphere: V = (4/3)πr³.
- Cavalieri's Principle: two solids with same height and cross-sectional areas have equal volume.
- Compare: cone = 1/3 cylinder; pyramid = 1/3 prism.
Cone r=3, h=4: V = (1/3)π(9)(4) = 12π. Sphere r=3: V = (4/3)π(27) = 36π.
Practice — Lessons 9–11
Q7 · Short Answer
Volume of a pyramid with square base 6 and height 9.
Q8 · Multiple Choice
How does the volume of a cone compare to a cylinder with the same base and height?
Topic C: Surface Area & Modeling
Lessons 12–18 · Surface area formulas and real-world applications.
12–15 — Lessons 12–15
Surface Area of 3D Figures
- Prism SA: 2(Base Area) + Lateral Area = 2B + Ph (P = perimeter of base).
- Cylinder SA: 2πr² + 2πrh.
- Pyramid SA: Base Area + (1/2)·Perimeter·slant height = B + (1/2)Pl.
- Cone SA: πr² + πrl (l = slant height).
- Sphere SA: 4πr².
Cylinder r=4, h=7: SA = 2π(16) + 2π(4)(7) = 32π + 56π = 88π.
Practice — Lessons 12–15
Q9 · Multiple Choice
Surface area of a sphere with radius 6 is:
Q10 · Short Answer
Total SA of a cube with side 5.
16–18 — Lessons 16–18
Modeling & Applications
- Real-world problems involve choosing the correct formula based on context.
- Painting a room → surface area. Filling a pool → volume.
- When combining shapes: calculate separately, then add or subtract as appropriate.
- Density: mass = density × volume. Find mass, volume, or density.
- Convert units carefully: 1 ft = 12 in, 1 m = 100 cm, etc.
A cylindrical tank (r=2 m, h=5 m) needs paint. SA = 2π(4) + 2π(2)(5) = 8π + 20π = 28π m².
Practice — Lessons 16–18
Q11 · Multiple Choice
You need to fill a rectangular pool 8m × 4m × 2m with water. You need to find:
Q12 · Multiple Choice
A composite solid is a cylinder on top of a cone. To find total volume: