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
- 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 (which is a rectangle) of a cylinder, A, with base radius, r, and height, h, is therefore given by
- This is the circumference of the circle, multiplied by the height
- This formula is not given to you in the exam
- The total surface area of a cylinder, ATotal, can be found using the formula
- This is the area of the curved surface (a rectangle), plus two circles of radius r
- 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 if it is needed
- The total surface area of a cone, ATotal, can be found using the formula
- This formula is not given to you in the exam
- It is just the curved surface area formula above, plus the area of a circle
- 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 if it is needed
Examiner Tip
- 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
Find the surface area of the cuboid shown below.
Find the area of the face at the front
2 cm × 10 cm = 20 cm2
Find the area of the face at the side
2 cm × 15 cm = 30 cm2
Find the area of the face at the top
10 cm × 15 cm = 150 cm2
There are two of each face
Add together the areas of all 6 faces
20 + 20 + 30 + 30 + 150 + 150 = 400
400 cm2