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 Pages
Surface Area of Cuboids
Surface Area of Prisms
Surface Area of a Sphere
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.
A cube is a three-dimensional figure with six equal square sides. The figure below shows a cube with sides s.
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}
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 = 6s^{2} where s is the length of a side.
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 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.
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
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
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}
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 terms of π)
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)
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.
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 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 + πr^{2}
= 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)
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.
A pyramid is named after the shape of its base. A rectangular pyramid has a rectangle base. A triangular pyramid has a triangle 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.
Surface area of regular pyramid = area of base + 1/2 ps
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 = b^{2} + 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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