Cones & spheres

Skills and methods in this topic, easy to hard

1. 1Volume of a pyramidV = 13 × base area × height

   Method13 × 6 × 5 × 8 = 80 cm³

   Another way: The prism, then ÷ 330 × 8 = 240 → 240 ÷ 3 = 80

2. 2Volume of a coneV = 13 πr²h

   Methodr = 3, h = 10: 13 × 9π × 10 = 30π cm³

   Another way: A third of the cylinder90π ÷ 3 = 30π

3. 3Surface area of pyramids and conesCone's curved surface πrl; l² = r² + h²

   Methodr = 5, h = 12: l = 13, 65π + 25π = 90π

   Another way: Open it into a sector513 × 169π = 65π

4. 4Volume of a sphereV = 43 πr³; hemisphere 23 πr³

   Methodr = 3: 43 × 27π = 36π cm³

   Another way: 2/3 of the cylinder23 × 54π = 36π

5. 5Surface area of a sphere4πr²; a solid hemisphere 3πr²

   Methodr = 5: 4π × 25 = 100π cm²

   Another way: The cylinder's curved side2π × 5 × 10 = 100π

6. 6Similar figures and solidsLengths a : b → areas a² : b² → volumes a³ : b³

   MethodHeights 2 : 3: volume 16 × 278 = 54 cm³

   Another way: The scale factor k16 × (32)³ = 54

7. 7Solids at workRecasting keeps the volume; frustum = big cone − small cone

   Method6³ ÷ 2³ = 27 balls

   Another way: A frustum by similarityHeights 1 : 2 → volumes 1 : 8 → 78

8. 8Folding a boxV = x(length − 2x)(width − 2x); a table finds the greatest

   Method20 × 16, cut 3: 3 × 14 × 10 = 420 cm³

   Another way: Draw V against xRead off the curve's highest point

Worked examples

From different skills in this topic. Have a go first, then open each one for the answer and the working.

1. Example 1Volume of a pyramid →

   The pyramid in the picture stands on a square of side 9 cm and is 4 cm high. What is its volume?

   • A. 324 cm³
   • B. 108 cm³
   • C. 162 cm³
   • D. 48 cm³

   Choose the right answer

   ▶Show the answer and working

   Answer: B. 108 cm³

   13 × 9² × 4 = 108 cm³.

2. Example 2Surface area of a sphere →

   Taking π as 3.14, what is the surface area of the sphere in the picture (radius 5 cm), in cm²?

   ▶Show the answer and working

   Answer: 314 cm²

   4 × 3.14 × 5² = 314 cm².

3. Example 3Folding a box →

   A 27 cm × 22 cm sheet, as in the picture, has a square of side 1 cm cut from each corner and is folded into a box with no lid. What is the box's volume, in cm³?

   ▶Show the answer and working

   Answer: 500 cm³

   The base is (27 − 2 × 1) × (22 − 2 × 1) = 25 × 20, the height 1: 500 cm³.