A cone is a three-dimensional shape with one circular base and one curved side that comes to a point called the vertex. Finding its surface area tells you how much material would cover the outside of the cone. This matters in real objects such as party hats, funnels, ice cream cones, and traffic cones.
The total surface area includes both the flat circular base and the curved lateral surface.
Key Facts
- Total surface area of a cone: SA = πr^2 + πrl
- Lateral surface area of a cone: LA = πrl
- Base area of a cone: B = πr^2
- Slant height formula: l = √(r^2 + h^2)
- The slant height l is measured along the side of the cone, not straight down the center.
- When the lateral surface is unwrapped, it forms a sector of a circle with arc length 2πr.
Vocabulary
- Cone
- A cone is a three-dimensional solid with a circular base and a curved surface that meets at one vertex.
- Radius
- The radius is the distance from the center of the circular base to the edge of the base.
- Height
- The height is the perpendicular distance from the center of the base to the vertex.
- Slant Height
- The slant height is the distance from the edge of the base to the vertex along the curved side of the cone.
- Lateral Surface Area
- Lateral surface area is the area of the curved side of the cone, not including the base.
Common Mistakes to Avoid
- Using h instead of l in SA = πr^2 + πrl is wrong because the curved surface depends on the slant height, not the vertical height.
- Forgetting the base area is wrong when total surface area is requested because total surface area includes πr^2 plus the lateral area.
- Doubling the radius incorrectly is wrong because the formula uses r, not diameter, unless you first divide the diameter by 2.
- Rounding too early is wrong because it can make the final answer less accurate, so keep extra digits until the last step.
Practice Questions
- 1 A cone has radius r = 4 cm and slant height l = 9 cm. Find its lateral surface area and total surface area in terms of π.
- 2 A cone has radius r = 6 m and height h = 8 m. Find the slant height, then find the total surface area in terms of π.
- 3 A student says the surface area of a cone is only πrl because the side is the largest visible part. Explain what part of the cone is missing from this calculation and when πrl alone would be appropriate.