The right circular cone

An ice-cream cone, a birth­day cap, a heap of sand and a tent are all shaped like a cone: a cir­cu­lar base that nar­rows to a point. This les­son intro­duces the three mea­sure­ments of a right cir­cu­lar cone, the radius, the height and the slant height, and gives its sur­face area and vol­ume. One sur­pris­ing fact appears along the way: a cone holds exactly one third of the cylin­der with the same base and height.

Parts of a cone#

A right cir­cu­lar cone has a cir­cu­lar base of radius $r$ and a ver­tex directly above the cen­tre of the base.

• The height $h$ is the per­pen­dic­u­lar dis­tance from the ver­tex to the cen­tre of the base.
• The slant height $l$ is the dis­tance from the ver­tex to any point on the edge of the base.

The radius, height and slant height form a right tri­an­gle with $l$ as hypotenuse, so

$$l = \sqrt{r^2 + h^2}$$

Sur­face area#

If the curved sur­face of a cone is cut along a slant line and laid flat, it becomes a sec­tor of a cir­cle of radius $l$. Its area works out to

$$\text{Curved surface area} = \pi r l$$

Adding the cir­cu­lar base,

$$\text{Total surface area} = \pi r l + \pi r^2 = \pi r(l + r)$$

Vol­ume#

$$\displaystyle \text{Volume of cone} = \frac{1}{3}\pi r^2 h$$

Com­pare it with a cylin­der of the same base radius and height, whose vol­ume is $\pi r^2 h$. Fill­ing the cone with water and pour­ing it into the cylin­der three times fills the cylin­der exactly.

Exam­ple 1#

A cone has radius $7$ cm and height $24$ cm. Find its slant height, curved sur­face area, total sur­face area and vol­ume.

$$l = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \text{ cm}$$

$$\displaystyle \text{CSA} = \frac{22}{7} \times 7 \times 25 = 550 \text{ cm}^2$$

$$\displaystyle \text{TSA} = 550 + \frac{22}{7} \times 7 \times 7 = 550 + 154 = 704 \text{ cm}^2$$

$$\displaystyle V = \frac{1}{3} \times \frac{22}{7} \times 7 \times 7 \times 24 = 1232 \text{ cm}^3$$

Exam­ple 2#

A con­i­cal tent has base radius $21$ m and height $20$ m. How much can­vas is needed to make it (ignor­ing the floor)?

$$l = \sqrt{21^2 + 20^2} = \sqrt{441 + 400} = \sqrt{841} = 29 \text{ m}$$

$$\displaystyle \text{Canvas} = \pi r l = \frac{22}{7} \times 21 \times 29 = 1914 \text{ m}^2$$

A tent has no floor of can­vas, so only the curved sur­face counts.

Exam­ple 3#

A cone has vol­ume $9856$ cm³ and base diam­e­ter $28$ cm. Find its height.

The radius is $14$ cm.

$$\displaystyle \frac{1}{3} \times \frac{22}{7} \times 14 \times 14 \times h = 9856 \implies \frac{616}{3}h = 9856 \implies h = \frac{9856 \times 3}{616} = 48 \text{ cm}$$

Exam­ple 4#

The curved sur­face area of a cone is $4070$ cm² and its base diam­e­ter is $70$ cm. Find its slant height and its height.

The radius is $35$ cm.

$$\displaystyle \frac{22}{7} \times 35 \times l = 4070 \implies 110l = 4070 \implies l = 37 \text{ cm}$$

$$h = \sqrt{37^2 - 35^2} = \sqrt{(37 - 35)(37 + 35)} = \sqrt{2 \times 72} = \sqrt{144} = 12 \text{ cm}$$

Writ­ing $37^2 - 35^2$ as a dif­fer­ence of squares saves a lot of mul­ti­pli­ca­tion.

Prac­tice#

Use $\displaystyle \pi = \frac{22}{7}$ unless told oth­er­wise.

1. A cone has radius $6$ cm and height $8$ cm. Find its slant height.
2. Find the curved sur­face area of a cone with radius $7$ cm and slant height $10$ cm.
3. Find the total sur­face area of a cone with radius $14$ cm and slant height $20$ cm.
4. Find the vol­ume of a cone with radius $3.5$ cm and height $12$ cm.
5. A cone has radius $21$ cm and height $28$ cm. Find its slant height, curved sur­face area and vol­ume.
6. A cone of radius $10$ cm has vol­ume $1570$ cm³. Find its height. (Use $\pi = 3.14$.)
7. A con­i­cal heap of wheat has base diam­e­ter $10.5$ m and height $3$ m. Find its vol­ume.
8. A cone and a cylin­der have the same base radius and the same height. The cylin­der's vol­ume is $360$ cm³. Find the cone's vol­ume.

Answers#