Calculating The Surface Area Of Cones

🔑 Key Concepts

  • The total surface area of a cone includes the lateral area and the circular base area.
  • The cone surface area formula is \( SA = \pi r\ell + \pi r^2 \).
  • The radius \( r \) is used in both the lateral area and base area parts of the formula.
  • The slant height \( \ell \) may be given directly or found using the Pythagorean theorem.
  • Surface area is measured in square units because it measures two-dimensional covering on a three-dimensional figure.

✏️ Worked Examples

🧠 Math Vocabulary

  • Surface Area: the total area covering the outside of a three-dimensional figure.
  • Lateral Area: the curved side area of a cone, represented by \( \pi r\ell \).
  • Base Area: the area of the circular base, represented by \( \pi r^2 \).
  • Radius: the distance from the center of the circular base to the edge of the circle.
  • Slant Height: the distance from the vertex of the cone to a point on the edge of the circular base.
  • Vertical Height: the perpendicular distance from the vertex of the cone to the center of the base.
  • Pythagorean Theorem: the relationship \( a^2 + b^2 = c^2 \), used to find missing side lengths in right triangles.
  • Circumference: the distance around a circle, which helps explain how the curved surface of a cone is formed.
  • Sector Area: part of the area of a circle; the lateral surface of a cone can be connected to a circular sector.
  • Square Units: units used to measure area, such as \( \text{cm}^2 \), \( \text{in}^2 \), or \( \text{ft}^2 \).

💡 Main Idea

To find the total surface area of a cone, add the lateral area and the area of the circular base. If the slant height is not given, use the radius, vertical height, and the Pythagorean theorem to find it before substituting into the surface area formula.

📚 What You Should Already Know

Students should already understand area of circles, using formulas, substituting values, evaluating powers, multiplying decimals, and applying the Pythagorean theorem to right triangles.

🚀 What Comes Next

After learning surface area of cones, students can compare formulas for cylinders, pyramids, prisms, and spheres, and solve real-world problems involving three-dimensional figures.

🧩 Embed This Video in Your LMS

Teachers can embed this YouTube video directly into Canvas, Google Classroom, Schoology, or another learning platform. Click below to show the embed code.