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

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}

This video shows how to find the surface area of a cube using the formula

Total surface area = 6

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)

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

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

How to find the surface area of a triangular prism by adding the area of the external faces?

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}

How to find the surface area of a sphere?

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 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.

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)

How to find the surface area of a cone?

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.

How to find the surface area of regular pyramid?

You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.

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