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Surface Area Formulas


Related Topics: More Geometry Lessons

In these lessons, we shall be looking at
  • a table of surface area formulas and volume formulas used to calculate the surface area and volume of three-dimensional geometrical shapes: cube, cuboid, prism, solid cylinder, hollow cylinder, cone, pyramid, sphere and hemisphere.
  • a more detailed explanation (in text and video) of each surface area formula.



 
The following table gives the surface area formulas for solid shapes or three-dimensional shapes. Scroll down the page if you need more explanations about the formulas, how to use them as well as worksheets.

surface area formulas



Surface Area of a Cube

A cube is a three-dimensional figure with six equal square sides. The figure below shows a cube with sides s.

surface area of cube

If s is the length of one of its sides, then the area of each side of a cube is s2. Since a cube has six square-shape sides, its total surface area is 6 times s2.
Surface area of a cube = 6s2

Worksheets and More Examples:
Worksheets: Volume & Surface Area of Cubes | More examples on the Volume of Cubes | More examples on the Surface Area of Cubes

How to find the surface area of a cube using the formula?
Total surface area = 6s2 where s is the length of a side
Example:
Given a side of 3 cm, find the surface area of the cube.

Rectangular Solid or Cuboids

A rectangular solid is also called a rectangular prism or a cuboid. In a rectangular solid, the length, width and height may be of different lengths.

surface area rectangular solid

The surface area of the above cuboid would be the sum of the area of all the surfaces which are rectangles.
Total area of top and bottom surfaces is lw + lw = 2lw
Total area of front and back surfaces is lh + lh = 2lh
Total area of the two side surfaces is wh + wh = 2wh

Surface area of rectangular solid = 2lw + 2lh + 2wh = 2(lw + lh + wh)

Worksheets and More Examples on Rectangular Prisms:
Worksheets: Volume & Surface Area of Cuboids | More examples on the Volume of Cuboids | More examples on the Surface Area of Cuboids

How to find the surface area of a rectangular prism or cuboid?
Example:
Find the surface of a rectangular prism with sides 18ft, 15ft and 20ft.



Surface Area of Prism

A prism is a solid that has two parallel faces which are congruent polygons at both ends. These faces form the bases of the prism. The other faces are in the shape of rectangles. They are called lateral faces. A prism is named after the shape of its base.

surface area prism

The surface area of a prism is the sum of the area of all its external faces.

We can also use the formula

Surface area of prism = 2 × area of base + perimeter of base × height

Worksheets and More Examples:
Worksheets: Volume of Prisms & Pyramids | More examples on the Volume of Prisms | More examples on the Surface Area of Prisms

How to find the surface area of a triangular prism by adding the area of the external faces? How to find the surface area of a triangular prism using the formula SA = ab+(s1+s2+s3)h?
where
a = altitude (height of the triangular face)
b = base of triangle
h = height of prism or distance between the two triangular faces.
s1, s2 and s3 are the three sides of the triangle

Surface Area of Sphere

A sphere is a solid in which all the points on the round surface are equidistant from a fixed point, known as the center of the sphere. The distance from the center to the surface is the radius.

surface area sphere

Surface area of a sphere with radius r = 4 πr2

Worksheets and More Examples:
Worksheets: Volume of Spheres | Worksheets: Surface Area of Spheres | More examples on the Volume of Spheres | More examples on the Surface Area of Spheres


How to find the surface area of a sphere?
Example:
Find the surface area of a sphere with r = 4ft. (leave answer in π form)

Surface Area of Solid Cylinder

A cylinder is a solid that has two parallel faces which are congruent circles. These faces form the bases of the cylinder. The cylinder has one curved surface. The height of the cylinder is the perpendicular distance between the two bases.

The net of a solid cylinder consists of 2 circles and one rectangle. The curved surface opens up to form a rectangle.

surface area cylinder

Surface area = 2 × area of circle + area of rectangle

Surface Area = 2πr2 + 2πrh = 2πr (r + h)

Worksheets and More Examples:
Worksheets: Volume of Cylinders
Worksheets: Surface Area of Cylinders
Worksheets: Volume & S.A. of Cylinders
Worksheets: Surface Area of Cylinders & Pipes
More examples on the Volume of Cylinders
More examples on the Surface Area of Cylinders

How to find the surface area of a cylinder?
How to obtain the total surface of a cylinder by looking at the net of the cylinder?
Example:
Find the surface area of a cylinder with radius 5 and height 12.

Surface area of hollow cylinder

Sometimes you may be required to calculate the total surface area of a hollow cylinder or tube.

surface area hollow cylinder

Total surface area of hollow cylinder
= area of internal curved surface + area of external curved surface + area of the two rings

= 2πrh + 2πRh + 2(πR2 − πr2)


Surface Area of Cone

A cone is a solid with a circular base. It has a curved surface which tapers (i.e. decreases in size) to a vertex at the top. The height of the cone is the perpendicular distance from the base to the vertex.

The net of a solid cone consists of a small circle and a sector of a larger circle. The arc of the sector has the same length as the circumference of the smaller circle.

Surface area of cone = Area of sector + area of circle
= πrs + πr2 = πr(r + s)

Worksheets and More Examples:
Worksheets: Volume of Cones | More examples on the Volume of Cones | More examples on the Surface Area of Cones

How to find the surface area of a cone?
Example:
Find the surface area of a cone with radius = 9cm, vertical height = 12cm and slant height = 15cm. (Leave answer in π form)

Surface area of Pyramid

A pyramid is a solid with a polygonal base and several triangular lateral faces. The lateral faces meet at a common vertex. The height of the pyramid is the perpendicular distance from the base to the vertex. The pyramid is named after the shape of its base.

surface area pyramid

We can find the surface area of any pyramid by adding up the areas of its lateral faces and its base.

Surface area of any pyramid = area of base + area of each of the lateral faces

If the pyramid is a regular pyramid, we can use the formula for the surface area of a regular pyramid.

Surface area of regular pyramid = area of base + halfps

where p is the perimeter of the base and s is the slant height.

If the pyramid is a square pyramid, we can use the formula for the surface area of a square pyramid.

Surface area of square pyramid = b2 + 2bs

where b is the length of the base and s is the slant height.

Worksheets and More Examples on Rectangular Prisms:
Worksheets: Volume of Square Pyramids
Worksheets: Volume of Prisms & Pyramids
More examples on the Volume of Pyramids
More examples on the Surface Area of Pyramids

How to find the surface area of regular pyramid?
Example:
Find the surface area of a regular pyramid with side = 40in, height = 39in and slant height = 44in.

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