A cone is a unique three-dimensional shape with a flat circular face at one end and a pointed tip at another end. The word ‘cone’ is derived from the Greek word ‘konos’, meaning a peak or a wedge.
A traffic signal cone, an ice-cream cone, or a birthday hat are some common examples of a cone.
Cone
Its circular face is the base.
Above the circular base is the curved surface that narrows to a pointed tip called the vertex (or apex).
Has no edge.
Since it has a curved surface, it is not a polyhedron.
The shape is achieved by rotating a triangle. So it can be said to be a rotated triangle.
Parts of a Cone
Parts of a Cone
Radius (r) – It is the distance between the center of its circular base to any point on the circumference of the base.
Height (h) – It is the perpendicular distance between its vertex to the base center. We consider the height of a right circular cone as its axis.
Slant Height (s) – It is the distance from its vertex to the point on the outer edge of its circular base.
Types of Cones – Right vs. Oblique
Based on the position of the vertex with respect to its base, a cone is of two types, as shown in the figure.
Right Circular Cone vs Oblique Cone
Right Circular Cone
It has its vertex aligned right above the center of the base. The axis coincides with the height and makes a right angle at the base center.
Oblique Cone
It does not have its vertex aligned perpendicular to its base. It looks tilted.
Since, in practical life, a cone means a right circular cone, here, we will learn the formulas related to it.
Formulas
Surface Area of a Cone
The formula of the surface area (or total surface area) of a right circular cone is:
Surface Area (SA) = πr2 + πrs, here r = radius, s = slant height, π = 3.141, πr2 = base area, πrs = lateral (curved) surface area of a cone (LSA)
So, we can rewrite the formula as SA = πr2 + LSA
Find the surface area of a right circular cone with a slant height of 6 cm and radius of 4 cm.
Solution:
As we know, Surface Area (SA) = πr2 + πrs, here r = 4 cm, s = 6 cm, π = 3.141 ∴ SA = 3.141 × 42 + 3.141 × 4 × 6 = 125.7 cm2
Calculate the lateral surface area of a cone with a radius of 3 in and slant height of 8 in.
Solution:
As we know, Lateral Surface Area (LSA) = πrs, here r = 3 in, s = 8 in, π = 3.141 ∴ LSA = 3.141 × 3 × 8 = 75.3 cm2
Find the base area of a cone with a radius of 7.5 cm.
Solution:
As we know, Base area = πr2, here π = 3.141, r = 7.5 cm = 3.141 × (7.5)2 = 176.7 cm2
Volume
The formula is:
Volume ${\left( V\right) =\dfrac{1}{3}\pi r^{2}h}$, here r = radius, h = height, π = 3.141
[Volume (V) = 1/3 πr2h]
We can relate the formula of the volume of a cylinder which is πr2h
Therefore, the volume of a cone is precisely one-third of that of a cylinder.
Calculate the volume of a cone with a radius of 7 mm and height of 12 mm.
Solution:
As we know, Volume ${\left( V\right) =\dfrac{1}{3}\pi r^{2}h}$, here r = 7 mm, h = 12mm, π = 3.141 ∴ ${V=\dfrac{1}{3}\times \pi \times 7^{2}\times 12}$ = 615.7 mm3
Slant Height
The formula is:
Slant Height ${\left( s\right) =\sqrt{r^{2}+h^{2}}}$, here r = radius, h = height
Find the slant height of a cone with a radius of 6 mm and height of 11 mm.
Solution:
As we know, Slant Height ${\left( s\right) =\sqrt{r^{2}+h^{2}}}$, here r = 6 mm, h = 11 mm ${\therefore s=\sqrt{6^{2}+11^{2}}}$ = 12.53 mm
Finding the HEIGHT of a cone when the VOLUME and RADIUS are known
Find the height of a cone with a volume of 65.9 mm3 and a radius of 3 mm..
Solution:
∵Volume ${\left( V\right) =\dfrac{1}{3}\pi r^{2}h}$, here r = radius, h = height, π = 3.141 So, by rearranging the equation, we get, Height ${h=\dfrac{3V}{\pi r^{2}}}$, here V = 65.9 mm3, r = 3 mm ${= \dfrac{3\times 65\cdot 9}{3.141\times 3^{2}}}$ = 6.99 mm
Finding the RADIUS when the VOLUMEand HEIGHT are known
Find the radius of a cone with a volume of 165.4cm2 and a height of 7.8 mm.
Solution:
∵Volume ${\left( V\right) =\dfrac{1}{3}\pi r^{2}h}$, here r = radius, h = height, π = 3.141 So, by rearranging the equation, we get, Radius ${\left( r\right) =\sqrt{\dfrac{3V}{\pi h}}}$, here V = 165.4 cm2, h = 7.8 cm ${= \sqrt{\dfrac{3\times 165\cdot 4}{3\cdot 141\times 7\cdot 8}}}$ = 4.5 cm
