Cone Volume and Surface Area

• A cone is a 3D object with a circular base narrowing to a single point (apex).
• Key cone measurements: radius (r), height (h), and slant height (l).

Cone Dimensions

• Radius (r): distance from the center of the circular base to its perimeter.
• Height (h): distance from the center of the base to the apex, inside the cone.
• Slant height (l): distance from the base perimeter to the apex, at an angle to the height.
• The height, radius, and slant height form a right triangle in a 3D cone.

Volume vs. Surface Area

• Volume: 3D space a cone occupies, measured in cubic units.
• Surface area: 2D space to wrap around the cone, measured in square units.

Surface Area

• The surface area (sa) of a cone: sa = πr(r + l).

Finding Volume

• The volume of a cone is related to its radius, height, and π (pi ≈ 3.14).
• Volume equation: v = (1/3)bh where v=volume, b=base area, h=height.
• First, find the area of the cone's base.

Area of the Base

• Cone's base is typically a circle.
• Area of a circle: A = πr².

Volume Formula

• The cone's volume is 1/3 of a cylinder with the same base/height.
• Volume = (1/3) * (base area) * (height).
• Volume is reported in cubed units.

Example Calculation

• If a cone has a base area of 28.26 cm² and a height of 7 cm, then the volume = (1/3) * 28.26 cm² * 7 cm = 65.94 cm³.

Example Ice Cream Cone

• Cone with a base radius of 1.5 inches and a height of 4 inches.
• Base area = π * (1.5 in)² = 7.07 in².
• Volume = (1/3) * 7.07 in² * 4 in = 9.42 in³.

Special Cones

• Standard cone: circular base, height forms a right angle with the radius.
• Irregular cones: oblique cones and frustums.

Oblique Cones

• Oblique cone: the apex does not align with the center of the base; the cone appears slanted.
• The volume equation is the same as for standard cones.
• Height: a line from the apex to the base forms a right angle, even if outside the cone's center.

Frustum

• Frustum: a cone with the tip cut off, resulting in two flat, circular surfaces
• R = larger radius, r = smaller radius, h = height
• Volume of frustum = (1/3)πh *(R² + Rr + r² )