Edexcel IGCSE Maths 复习笔记 4.15.1 3D Shapes - Surface Area

What is surface area?

• A face is one of the flat or curved surfaces that make up a 3D shape
• The surface area of a 3D shape is the sum of the areas of all the faces that make up the shape
• Note how we are carrying a 2D idea (area) into 3 dimensions here

Surface area – cuboids, pyramids, and prisms

• In cuboids, polygonal-based pyramids, and polygonal-based prisms (ie. pyramids and prisms whose bases have straight sides), all the faces are flat
• The surface area is found simply by adding up the areas of these flat faces

• When calculating surface area, it can be very helpful to draw a 2D net for the 3D shape in question
• For example:
  • The base of a square-based pyramid is 15 cm on a side
  • The triangular faces are identical isosceles triangles, each with a height (from the base to the top of the pyramid) of 23 cm
  • Find the total surface area of the pyramid
  • Draw a net for the shape:

   

Area of square base =15² = 225 cm²

Area of one triangular face = ½ base × height = ½ × 15 × 23 =172.5 cm²

Total surface area =225 + 4 × 172.5 = 915 cm²

Surface area – cylinders, cones, and spheres

• All three of these shapes have curved faces, so we have to be a little more careful when calculating their surface areas

1. The net of a cylinder consists of two circles and a rectangle: 

• The total surface area of a cylinder with base radius r and height h is therefore given by:Total surface area of a cylinder= 2πr² + 2πrh

2. The net of a cone consists of the circular base along with the curved surface area:

• The length l in that diagram is known as the slant height (while h is the vertical height of the cone)
• To find the surface area of a cone with base radius r and slant height l, we use the formulas:Curved surface area of a cone = πrlTotal surface area of a cone = πr²+πrl

3. To find the surface area of a sphere with radius r, use the formula:

Surface area of a sphere=4πr²