The surface area is the area that describes the material that will be used to cover a geometric solid. When we determine the surface areas of a geometric solid we take the sum of the area for each geometric form within the solid.

The volume is a measure of how much a figure can hold and is measured in cubic units.

The volume tells us something about the capacity of a figure.

A prism is a solid figure that has two parallel congruent sides that are called bases that are connected by the lateral faces that are parallelograms. There are both rectangular and triangular prisms.

To find the surface area of a prism (or any other geometric solid) we open the solid like a carton box and flatten it out to find all included geometric forms.

To find the volume of a prism (it doesn't matter if it is rectangular or triangular) we multiply the area of the base, called the base area B, by the height h.

V • B h

A cylinder is a tube and is composed of two parallel congruent circles and a rectangle which base is the circumference of the circle.

Example

The area of one circle is: The circumference of a circle: The area of the rectangle:

A - πr2

A - tt • 22

A - tt • 4

A = 12.6

C = πD

C = π • 4

C = 12.6

A = C • H

A = 12.6 • 6

A = 75.6

The surface area of the whole cylinder:

A = 75.6 + 12.6 + 12.6 = 100.8 units2

To find the volume of a cylinder we multiply the base area (which is a circle) and the

height h.

V - nr2 • h

A pyramid consists of three or four triangular lateral surfaces and a three or four sided surface, respectively, at its base. When we calculate the surface area of the pyramid below we take the sum of the areas of the 4 triangles area and the base square. The height of a triangle within a pyramid is called the slant height.

The volume of a pyramid is one third of the volume of a prism.

V = 1/3 • B • h

The base of a cone is a circle and that is easy to see. The lateral surface of a cone is a parallelogram with a base that is half the circumference of the cone and with the slant height as the height. This can be a little bit trickier to see, but if you cut the lateral surface of the cone into sections and lay them next to each other it's easily seen.

The surface area of a cone is thus the sum of the areas of the base and the lateral surface:

Abase = πr2 and Als = πrl

A = πr2 + πrl

Example

Abase = πr2

Abase = π • 32

Abase = 28.3

Als = πrl

Als = π • 3 • 9

Als = 84.8

A - πr2 + πrl = 28.3 + 84.8 = 113.1 units2

The volume of a cone is one third of the volume of a cylinder.

V = 1/3 π • r2 • h

Find the volume of a prism that has the base 5 and the height 3.

B - 3 • 5 = 15

V = 15 • 3 = 45 units3

Video lesson: Find the surface area of a cylinder with the radius 4 and height 8.

To see the solution look at the video in the next page.

For this problem let’s find the surface are of a cylinder. In a cylinder we’re given a radius of 4 and height of 8. To find the surface area of a cylinder we need to find the area of both the top and bottom circle and the rectangle is the piece that wraps all away round the circle. It’s basically a rectangle if you unwrap it. So to find the area of one circle it’s just πr2. So the area of both circles is just 2 times pi times r squared. So r squared is 4 times 4 or 16, so that comes out to be 32π which comes out to be approximately 100.6. The area for the rectangle that wraps around the circle is going to be the base times the height. So the height is going to be 8, the base is going to be the circumference of a circle. The circumference of a circle is 2πr. 2 times pi and r is 4. If you multiply all this out you get 16 times 4 time pi which comes out to be 201.1. Now to find the total surface area of a cylinder you have to add the 2 circles and the rectangle together. So 100.6 plus 201.1 which comes out to be, if you add those together, 301.7 is the approximate surface area of the whole cylinder.