What is the volume of a cone?
Answer: Volume of a cone = 13πr2h = 13 × area of base × height: exactly one-third of a cylinder with the same base and height.
Step-by-step solution
Idea: Compare a cone with a cylinder of the same base radius and height. Filling the cone with salt and pouring it into the cylinder, it takes exactly three conefuls to fill the cylinder.
- Take a cylindrical tin and make a paper cone with the same base radius r and the same height h.½ mark
- Fill the cone with fine salt and empty it into the cylinder. After one and two conefuls the cylinder is not full; after the third coneful it is full to the brim.1 mark
- So volume of cylinder = 3 × volume of cone, i.e. volume of cone = 13 × πr2h.1 mark
- In words: V = 13 × area of base × height. (A full proof of the 13 needs calculus, studied later.)½ mark
Volume of a cone = (1/3)πr²h = (1/3) × area of base × height.
Check: Cone with r = 3 cm, h = 7 cm: 13 × 227 × 9 × 7 = 66 cm3; cylinder: 227 × 9 × 7 = 198 cm3 = 3 × 66 ✓.
Answer to write in the exam
Volume of cylinder (same r, h) = 3 × volume of cone
∴ Volume of cone = 13πr2h = 13 × area of base × height
Common mistakes that cost marks
- Forgetting the 13 and using πr2h.
- Using the slant height l in place of the height h.
- Using the diameter instead of the radius in r2.
How this can come in the exam
MCQ (1 mark)
A cone and a cylinder have the same base and the same height. If the cylinder’s volume is 270 cm3, the cone’s volume is
- 810 cm3
- 135 cm3
- 90 cm3
- 270 cm3
Show answer
(C) 90 cm3
Cone = 13 × cylinder = 90 cm3.
Short answer (2 marks)
Find the volume of a cone of radius 7 cm and height 15 cm. (π = 227)
Show answer
13 × 227 × 49 × 15 = 22 × 7 × 5 = 770 cm3.Try one yourself
Find the volume of a cone with base area 154 cm2 and height 9 cm.
Show answer
13 × 154 × 9 = 462 cm3.
More questions like this
- Here is another hands-on activity that uses modelling clay (earlier this used to be called ‘plasticine’). Carefully mold the clay into a solid cylinder and measure its radius and height. Then reshape the same portion of clay into identical cones, each having the same radius and the same height as the cylinder. You will find that you are able to make exactly three such cones.
- A right triangle ABC with sides 5 cm, 12 cm and 13 cm is rotated through 360° about the side with length 12 cm. Find the volume of the solid so obtained.
- Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m.
- Find the curved surface area of a right circular cone whose slant height is 10 cm and base radius is 7 cm.
- The height of a cone is 16 cm, and its base radius is 12 cm. Find the curved surface area and the total surface area of the cone.