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Surface Area Formulas




 
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.
Related Topics: More Geometry Lessons

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. The dotted lines indicate edges hidden from your view.

surface area 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


This video shows how to find the surface area of a cube using the formula
Total surface area = 6s2 where s is the length of a side


 

Rectangular Solids 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.
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)


How to find the surface area of a rectangular prism or cuboid?


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


How to find the surface area of a triangular prism by adding the area of the external faces?
This video shows how to find the surface area of a triangular prism using the formula SA = ab+(s1+s2+s3)h.


 

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


How to find the surface area of a sphere?


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)


How to find the surface area of a cylinder? This video will show how to obtain the total surface of a cylinder by looking at the net of the cylinder.


 

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)

How to find the surface area of a cone?

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.


How to find the surface area of regular pyramid?

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.


You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.


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