Surface Area
What is surface area?
- The surface area of a 3D object is the sum of the areas of all the faces that make up the shape
- Area is a 2D idea being applied into a 3D situation
- A face is one of the flat or curved surfaces that make up a 3D object
How do I find the surface area of cubes, cuboids, pyramids, and prisms?
- In cubes, 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 by
- calculating the area of each individual flat face
- adding these areas together
- You should remember the formula for the area of a rectangle, but you are given the area of a triangle in your exam
- When calculating surface area, it can be helpful to draw a 2D net for the 3D shape in question
- For example, consider a square-based pyramid where the top of the pyramid is directly above the centre of the base
- Its net will consist of a square base and four identical isosceles triangular faces
- Calculate the area of a square and the area of each triangle then add them together
- For example, consider a square-based pyramid where the top of the pyramid is directly above the centre of the base
How do I find the surface area of a cylinder?
- A cylinder has two flat surfaces (the top and the base) and one curved surface
- The net of a cylinder consists of two circles and a rectangle
- The curved surface area of a cylinder, A, with base radius, r, and height, h, is therefore given by
- This formula is given to you in the exam
- The total surface area of a cylinder, ATotal, can be found using the formula
- This formula is not given to you in the exam
How do I find the surface area of a cone?
- A cone has one flat surface (the base) and one curved surface
- The net of a cone, with radius, r, perpendicular height, h, and sloping edge, (slant height), l, consists of
- A circular base
- A sector with radius, l, and an arc length equal to the circumference of the base
- The curved surface area of a cone, A, with radius, r, perpendicular height, h, and sloping edge, l, can be found using the formula
- This formula is given to you in the exam
- The total surface area of a cone, ATotal, can be found using the formula
-
- This formula is not given to you in the exam
How do I find the surface area of a sphere?
- A sphere has a single curved surface
- The surface area of a sphere, A, with radius, r, can be found using the formula
- This formula is given to you in the exam
- A hemisphere has half the curved surface area of a sphere and the flat circular base
- The surface are of a hemisphere, A, with radius, r, can be found using the formula
- This formula is not given to you in the exam
Examiner Tip
- The curved surface area for a cylinder, cone and sphere are given to you in the exam
- Read the question carefully, you may need to add additional areas, e.g. a base
- Make you are confident in calculating the areas of rectangles, circles and triangles
Worked example
A toy consists of a cone of radius 5 cm and slant height 12 cm placed on top of a hemisphere with the same radius. Find the total surface area of the toy. Give your answer to 3 significant figures.
Calculate the curved surface area of the cone using the formula,
Calculate the curved surface area of a hemisphere using the formula for the curved surface area of a sphere, and dividing it by 2
Add the two areas together to find the total surface area of the toy
Evaluate on your calc and round to 3 significant figures
346 cm2 (3 s.f.)