Pyramids (Triangular Pyramids and Cones)
The volume of a pointed solid figure (such as a triangular pyramid or a cone, called a pyramid) is 1/3 of the volume of a prism with the same base and the same height. Surface area involves the slant height.
Formulas for Pyramids
- Volume = base area × height × 1/3
- Lateral area of a cone = π × r × l （r is the radius of the base, and l is the slant height, the length from the vertex to the edge of the base）
- Surface area of a cone = π × r² + π × r × l
Example 1 (Volume of a Cone): Find the volume of a cone with a base radius of 4 cm and a height of 9 cm. (π=3.14)
Solution: (1/3) × 4² × 3.14 × 9 = (1/3) × 16 × 3.14 × 9 = 150.72 cm³
Solution: (1/3) × 4² × 3.14 × 9 = (1/3) × 16 × 3.14 × 9 = 150.72 cm³
Example 2 (Surface Area of a Cone): Find the surface area of a cone with a base radius of 3 cm and a slant height of 5 cm. (π=3.14)
Solution:
① Base area = 9 × 3.14 = 28.26 cm²
② Lateral area = 3.14 × 3 × 5 = 47.1 cm²
③ Surface area = 28.26 + 47.1 = 75.36 cm²
Solution:
① Base area = 9 × 3.14 = 28.26 cm²
② Lateral area = 3.14 × 3 × 5 = 47.1 cm²
③ Surface area = 28.26 + 47.1 = 75.36 cm²
The slant height l is the length of the straight line connecting the vertex of a cone to the edge of its circular base. In the net of a cone, the slant height is the radius of the sector.