Net of a Cone
Cone Net
See how a 3D cone unfolds into its flat net - a sector and a circle.
tap to unfold the cone
Relation Among Radius, Height and Slant Height
Relation Among Radius, Height
and Slant Height
A O B C h l r

Consider a cone with vertex A. Let OB = r cm be the radius of the base of the cone.

Join A to O (the centre of the base). Then, AO is the height of the cone, i.e., AO = h.

AC and AB are the slant heights of the cone, i.e., AC = AB = l.

In right triangle AOB, by Pythagoras Theorem:
→ AB2 = AO2 + OB2
→ l2 = h2 + r2
→ l = √(h2 + r2)

The slant height of a cone of radius r and height h is equal to √(h² + r²).

Surface Area of Cone
Surface Area of Cone

Take a cone made from a paper. Cut it as its slant surface and open it as shown in the Figure. Now, the curved surface of cone is transferred to the flat form as shown in the Figure, which is a sector. And the length of this sector is equal to the circumference of the circular base of the cone.

l l 2πr
Figure (i)

Now, cut the sector into 4 equal pieces as shown in the Figure (ii) and color two pieces with red.

Then, arrange two small sectors in the same directions and remaining two in opposite directions as shown in figure (iii).

l πr πr
Figure (iii)

Here,
If the radius of base of the cone is 'r', slant height is 'ℓ' and the vertical height is 'h', then,

Curved Surface Area (CSA) Derivation
Curved surface area of the cone (CSA) = Area of parallelogram
= base × height (Note: The parallelogram base is half the original arc length)
= πr × ℓ
= πrℓ sq. unit
Curved Surface Area (CSA)

Curved surface area of the cone (CSA) = πr² + πrℓ

Total Surface Area (TSA)

Total surface area of the cone (TSA) = Area of base + Curved surface area (CSA)

Total surface area (TSA) = πr² + πrℓ

Total surface area (TSA) = πr(r + ℓ) sq. unit

Volume of Cone

Experimental Verification: Volume of a Cone

Explore how the volume of a cone relates to a cylinder with the same base radius and height.

Cones Poured: 0 / 3

Activity Observation

Fill the cone with sand or dust of soil and pour it into the cylinder until it is filled up completely.

  1. How many times should it be poured from the cone to fill up the cylinder? 👉 It must be poured exactly 3 times.
  2. Is it filled in three times? 👉 Definitely, the cylinder is completely filled after pouring three times.
Mathematical Deduction:

Since 3 full cones fill exactly 1 cylinder of equal dimensions:

Volume of the Cone (V) = 1⁄3 × Volume of Cylinder
V = 1⁄3 × A × h = 1⁄3 π r2 h
[ Where Area of base (A) = π r2 ]
Solve Questions — Cone Calculator
CSA = πrl  |  TSA = πr(l+r)  |  V = ⅓πr²h  |  l = √(r²+h²)

🔺 Cone Problems
🎉 Excellent work! Well done!
Cone

A cone is a three-dimensional geometric shape formed by rotating a right-angled triangle around one of its sides (the axis).

It has a circular base and a pointed apex. The net of a cone consists of:

1. A circular base — the flat bottom of the cone
2. A circular sector — the curved lateral surface when unfolded

By adjusting the slider below, you can see how the cone folds and unfolds to reveal its net.

UNFOLD FOLD