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Surface Area (Edexcel IGCSE Maths)
Revision Note
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
How do I find the surface area of 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
How do I find the surface area of 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 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 formulae:
- Curved surface area of a cone = πrl
- Total 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
Examiner Tip
- The formula for the surface area of a sphere or the curved surface area of a cone will be given to you in an exam question if you need it
- The rest of the formulae here come from what you should already know about areas of rectangles, triangles, and circles
- Be careful when calculating the surface area of a hemisphere:The total surface area consists of the curved part (half of a sphere) PLUS the flat circular face – so the total surface area is 3πr2
Worked example
The base radius, , of a cone is the same as the radius of a hemisphere. The total surface area of the cone is equal to the total surface area of the hemisphere.
Remember that a hemisphere has both a curved surface area and a flat circular face so the formula for the surface area is:
Remember that a cone has both a curved surface area and a flat circular face so the formula for the surface area is:
GIve your answer accurate to 1 decimal place.
The first decimal place is a 2, and this is followed by a 2 so you do not need to round it up.
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