📚 Practice With Worksheets
Use these worksheets to practice setting up and calculating the surface area of cones.
▶️ 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
Use the cone diagram and right triangle view to identify the radius, height, and slant height.
Enter the radius and slant height into the surface area formula, then combine the terms with \(\pi\).
In hard mode, one dimension may be missing, so use the right triangle relationship before calculating surface area.
🧠 Things to Notice
The surface area formula uses the radius and slant height: \(SA=\pi rL+\pi r^2\).
The vertical height is not directly substituted into the surface area formula.
The radius, vertical height, and slant height form a right triangle, so \(r^2+h^2=L^2\).
📘 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, measured in square units.
📐 Cone Surface Area Concepts
The total surface area of a cone is the lateral area plus the circular base area.
\(SA=\pi rL+\pi r^2\)
When the slant height is missing, use the radius and vertical height as the legs of a right triangle.
\(r^2+h^2=L^2\)
The slant height \(L\) acts like the hypotenuse, while \(r\) and \(h\) act like the legs.
✏️ Worked Example
Find the surface area of a cone with radius \(r=6\) units and slant height \(L=10\) units.
1. Given
\(r=6\)
\(L=10\)
2. Lateral Area
\(LA=\pi rL\)
\(LA=\pi(6)(10)=60\pi\)
3. Base Area
\(B=\pi r^2\)
\(B=\pi(6)^2=36\pi\)
4. Total Surface Area
\(SA=60\pi+36\pi\)
\(SA=96\pi\approx301.59\)
When a Dimension Is Missing
If the slant height is not given, use the right triangle inside the cone. For example, if \(r=5\) and \(h=12\), then \(L\) is the hypotenuse.
\(r^2+h^2=L^2\)
\(5^2+12^2=L^2\)
\(25+144=L^2\)
\(169=L^2\)
\(L=13\)
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