Chapter Overview

What You'll Learn

Key Highlights

Chapter covers surface areas of cones (derived from net sector, \( l = \sqrt{r^2 + h^2} \)), spheres (4πr² from rotation), hemispheres, and combinations like cone on cylinder. Volumes include cone (1/3πr²h), sphere (4/3πr³). Applications: tents, caps, painting costs, grain estimation on cob.

Comprehensive Chapter Summary

1. Surface Area of a Right Circular Cone

• Recall: Studied surface areas of cube, cuboid, cylinder; now cone, a pyramid-like solid generated by rotating right triangle around perpendicular side.
• Activity: Cut right-angled triangle ABC (right at B), paste string on AB, rotate around string; forms cone shape like ice-cream cone (Fig. 11.1).
• Definitions: Vertex A, height AB = h, base radius BC = r, slant height AC = l; B center of base (Fig. 11.1(c)).
• Not right circular cone: If axis not perpendicular to base (Fig. 11.2(a)) or base not circular (Fig. 11.2(b)).
• Activity (i): Cut paper cone along slant height l, open; forms sector of circle like cake slice (Fig. 11.3(a)).
• Activity (ii): Bring tips A, B together; curved edge forms base circumference 2πr (Fig. 11.3(c)).
• Activity (iii): Cut sector into small triangles from O; each ≈ triangle with height l.
• Activity (iv): Area each triangle = \( \frac{1}{2} \times \) base × l; total area = \( \frac{1}{2} l \times \) (sum bases) = \( \frac{1}{2} l \times 2\pi r = \pi r l \).
• Curved Surface Area (CSA) Cone = \( \pi r l \), where r base radius, l slant height.
• Relation: \( l^2 = r^2 + h^2 \) (Pythagoras, Fig. 11.4); \( l = \sqrt{r^2 + h^2} \).
• Total Surface Area (TSA) = CSA + base area = \( \pi r l + \pi r^2 = \pi r (l + r) \).
• Example 1: l=10 cm, r=7 cm; CSA = \( \pi \times 7 \times 10 = 220 \) cm².
• Example 2: h=16 cm, r=12 cm (\( \pi=3.14 \)); l=20 cm; CSA=753.6 cm², TSA=1205.76 cm².
• Example 3: Corn cob r=2.1 cm, h=20 cm; l≈20.11 cm; CSA≈132.73 cm²; grains=4/cm² → ≈531 grains.
• Elaboration: Derivation approximates sector as triangles; exact for limit; useful for nets, unrolling surfaces.
• Application: Tents (slant for canvas), caps (sheet area), painting (outer surface).
• Exercise 11.1: Diameter 10.5 cm (r=5.25 cm), l=10 cm; CSA=165 cm²; TSA=231 cm²; etc.
• Extensions: Oblique cones differ, but focus right circular; combine with cylinders for real objects.
• Verification: Measure physical cone net; circumference matches base.

2. Surface Area of a Sphere

• Sphere: Solid from rotating semicircle around diameter; every point equidistant r from center.
• Activity: Paste string on circular disc diameter, rotate; forms sphere like ball (Fig. 11.6).
• Surface Area: Derived from surface of zone; total TSA = \( 4\pi r^2 \).
• Great Circle: Any plane through center intersects sphere in circle radius r.
• Hemisphere: Half sphere; curved SA = \( 2\pi r^2 \), TSA = \( 3\pi r^2 \) (includes base).
• Example: Sphere r=7 cm; TSA = \( 4 \times \frac{22}{7} \times 49 = 616 \) cm².
• Elaboration: Archimedes' method: Sphere SA equals cylinder lateral minus bases; intuitive from projection.
• Application: Balls, globes; painting/ coating costs.
• Relation to Cone: Sphere as limit of polyhedra; formulas unify curved surfaces.
• Verification: Compare with known volumes; SA/volume ratio for spheres minimal.
• Extensions: Spherical caps, zones for partial surfaces.

3. Combinations of Solids

• Composite: Cone on cylinder (ice-cream); SA = cylinder lateral + cone CSA (no bases if joined).
• Volume: Sum individual; e.g., cone + hemisphere = \( \frac{1}{3}\pi r^2 h + \frac{2}{3}\pi r^3 \).
• Example: Cylinder h=10 cm, r=3 cm + cone h=4 cm; combined TSA calculation excludes internal base.
• Activity: Build paper models; measure total SA.
• Elaboration: Subtract overlapping surfaces; add exposed.
• Application: Tanks, tents with poles; packaging costs.
• Exercise: Find SA of cone-sphere, volume conversions.
• Extensions: Frustum (cone slice) SA = π(r1+r2)l + πr1² + πr2².
• Verification: Compare calculated vs measured paint needed.

4. Volumes of Solids

• Cone Volume: \( V = \frac{1}{3} \pi r^2 h \); third cylinder same base/height.
• Sphere Volume: \( V = \frac{4}{3} \pi r^3 \); derived from integration or Cavalieri.
• Hemisphere: \( \frac{2}{3} \pi r^3 \).
• Combinations: Add/subtract; e.g., sphere in cone volume difference.
• Example: Cone h=15 cm, r=6 cm; V= \( \frac{1}{3} \times \frac{22}{7} \times 36 \times 15 = 660 \) cm³.
• Elaboration: Scaling: Volume cubes linear dimensions; SA squares.
• Application: Storage capacity, material displacement.
• Exercise: Convert units, find missing dimensions.
• Extensions: Frustum volume \( \frac{1}{3} \pi h (r_1^2 + r_2^2 + r_1 r_2) \).
• Verification: Water fill experiments match formulas.

Key Concepts and Definitions

Important Facts

Questions and Answers from Chapter

Short Questions (1 Mark)

Medium Questions (3 Marks)

Long Questions (6 Marks)

Quick Revision Notes