📚 Practice With Worksheets
Use these worksheets to practice finding surface area after exploring how the cone formula is connected to circles and sectors.
▶️ Learn With Videos
Watch the related lessons to learn how to calculate the surface area of cones and compare cone and sphere surface area formulas.
🛠️ How to Use This Tool
Move the radius and height sliders to change the cone and watch the slant height update.
Compare the cone, the unwrapped lateral surface, and the circular base to see how the pieces of surface area fit together.
Use the measurements and formula connection to explore why the total surface area is the lateral area plus the base area.
🧠 Things to Notice
The slant height is the hypotenuse of a right triangle formed by the radius and vertical height.
When the cone is unwrapped, the lateral surface becomes a sector of a circle.
The sector fraction is \(\frac{r}{L}\), which is why the lateral area simplifies to \(\pi rL\).
📘 Vocabulary
Cone: A three-dimensional figure with one circular base and one curved lateral surface that meets at a vertex.
Radius: The distance from the center of the circular base to the edge of the base.
Vertical height: The perpendicular distance from the base of the cone to the vertex.
Slant height: The distance from the vertex of the cone to the edge of the circular base.
Lateral area: The area of the curved side of the cone, not including the base.
Base area: The area of the circular base of the cone.
Total surface area: The lateral area plus the base area.
📐 Cone Surface Area Concepts
The slant height can be found using the Pythagorean theorem when the radius and vertical height are known.
\(L=\sqrt{r^2+h^2}\)
The lateral surface of a cone unwraps into a sector. The fraction of the full circle is the base radius divided by the slant height.
\(\text{sector fraction}=\frac{r}{L}\)
The total surface area of a cone is the lateral area plus the circular base area.
\(SA=\pi rL+\pi r^2\)
🧮 Worked Example
Find the surface area of a cone with radius \(r=3\) units and slant height \(L=6\) units.
1. Given
\(r=3\)
\(L=6\)
2. Lateral Area
\(LA=\pi rL\)
\(LA=\pi(3)(6)=18\pi\)
3. Base Area
\(B=\pi r^2\)
\(B=\pi(3)^2=9\pi\)
4. Total Surface Area
\(SA=18\pi+9\pi\)
\(SA=27\pi\approx84.82\)
Lateral Area Derivation
When the cone is unwrapped, the curved side becomes a sector of a circle. The radius of that larger circle is the slant height \(L\).
The arc length of the sector equals the circumference of the cone's base:
\(\text{arc length}=2\pi r\)
The full circumference of the larger circle is:
\(2\pi L\)
So the sector uses this fraction of the full circle:
\(\frac{2\pi r}{2\pi L}=\frac{r}{L}\)
Now multiply that fraction by the area of the larger circle:
\(\left(\frac{r}{L}\right)(\pi L^2)=\pi rL\)
That is why the lateral area formula for a cone is \(LA=\pi rL\).
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