In these lessons, we shall be looking at

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 of a Cube

**How to find the surface area of a cube using the formula?**

Total surface area = 6*s*^{2} where *s* is the length of a side

Example:

Given a side of 3 cm, find the surface area of the cube.

### Rectangular Solids or 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

**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

**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

**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

= area of internal curved surface + area of external curved surface + area of the two rings

### Surface Area of Cone

**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

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

**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.

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.

- 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.

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.

If *s* is the length of one of its sides,
then the area of each side of a cube is *s*^{2}.
Since a cube has six square-shape sides, its total surface area is
6 times *s*^{2}.

Surface area of a cube = 6s^{2}

Total surface area = 6

Example:

Given a side of 3 cm, find the surface area of the cube.

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.

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 =
*2*lw
*Total area of front and back surfaces is

Surface area of rectangular solid = 2

lw+ 2lh+ 2wh= 2(lw+lh+wh)

Example:

Find the surface of a rectangular prism with sides 18ft, 15ft and 20ft.

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.

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

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

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 of a sphere with radius

r= 4 πr^{2}

Example:

Find the surface area of a sphere with r = 4ft. (leave answer in π form)

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 = 2 × area of circle + area of rectangle

Surface Area = 2πr^{2}+ 2πrh= 2πr(r + h)

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.

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

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

= 2πrh+ 2πRh +2(πR^{2}− πr^{2})

A cone is a solid with a circular **bas****e**.
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+ πr^{2}= πr(r+s)

Example:

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

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.

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.

whereSurface area of regular pyramid = area of base +

ps

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 =

b^{2}+ 2bs

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

Example:

Find the surface area of a regular pyramid with side = 40in, height = 39in and slant height = 44in.

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.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.