# Appendix T: Fundamental Relationships from Geometry

OpenStax

This chapter is from 9.4, 9.5, and 9.6 in OpenStax PreAlgebra 2e.

# Rectangles

A rectangle has four sides and four right angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length and the adjacent side as the width, . See the figure below

The perimeter, of the rectangle is the distance around the rectangle. If you started at one corner and walked around the rectangle, you would walk

or

.

What about the area of a rectangle? Consider the rug shown below: it is 2 feet long by 3 feet wide, and its area is 6 square feet. Since , we see that the area, , is the length, , times the width, , so the area of a rectangle is

.

Properties of Rectangles

• Rectangles have four sides and four right (90°) angles.
• The lengths of opposite sides are equal.
• The perimeter, of a rectangle is the sum of twice the length and twice the width:
• The area, of a rectangle is the length times the width

# Triangles

We now know how to find the area of a rectangle. We can use this fact to help us visualize the formula for the area of a triangle. In the rectangle below, we’ve labeled the length and the width so it’s area is .

We can divide this rectangle into two triangles as shown below. Triangles that are congruent have identical side lengths and angles, and so their areas are equal. The area of each triangle is one-half the area of the rectangle, or . This example helps us see why the formula for the area of a triangle is .

The formula for the area of a triangle is where is the base and is the height.

To find the area of the triangle, you need to know its base and height. The base is the length of one side of the triangle, usually the side at the bottom. The height is the length of the line that connects the base to the opposite vertex, and makes a 90° angle with the base. The figure below shows three triangles with the base and height of each marked.

Triangle Properties

• For any triangle , the sum of the measures of the angles is : .
• The perimeter of a triangle is the sum of the lengths of the sides.
• The area of a triangle is one-half the base, b, times the height, h.

# Circles

Properties of Circles

• is the length of the radius: the distance from the center to the edge of the circle.
• is the length of the diameter: the distance across the circle through the center
• Circumference is the perimeter of a circle. The formula for circumference is
• The formula for area of a circle is

# Surface Areas and Volumes

## Rectangular Solids

### Volume

A cheer-leading coach is having the squad paint wooden crates with the school colors to stand on at the games. (See the figure below). The amount of paint needed to cover the outside of each box is the surface area, a square measure of the total area of all the sides. The amount of space inside the crate is the volume, a cubic measure.

Each crate is in the shape of a rectangular solid. Its dimensions are the length, width, and height. The rectangular solid shown below has length 4 units, width 2 units, and height 3 units. Can you tell how many cubic units there are altogether? Let’s look layer by layer.

Altogether there are 24 cubic units. Notice that 24 is the length×width×height.

The volume, of any rectangular solid is the product of the length, width, and height.

We could also write the formula for volume of a rectangular solid in terms of the area of the base. The area of the base, is equal to length×width.

We can substitute for in the volume formula to get another form of the volume formula.

We now have another version of the volume formula for rectangular solids. Let’s see how this works with the 4×2×3 rectangular solid we started with

### Surface Area

To find the surface area of a rectangular solid, think about finding the area of each of its faces. How many faces does the rectangular solid above have? You can see three of them.

Notice for each of the three faces you see, there is an identical opposite face that does not show.

The surface area S of the rectangular solid shown in below is 52 square units.

In general, to find the surface area of a rectangular solid, remember that each face is a rectangle, so its area is the product of its two dimensions, either length and width, length and height, or width and height (see the figure below). Find the area of each face that you see and then multiply each area by two to account for the face on the opposite side:

Volume and Surface Area of a Rectangular Solid

For a rectangular solid with length , width and height

## Spheres

A sphere is the shape of a basketball, like a three-dimensional circle. Just like a circle, the size of a sphere is determined by its radius, which is the distance from the center of the sphere to any point on its surface. The formulas for the volume and surface area of a sphere are given below.

Volume and Surface Area of a Sphere

For a sphere with radius r: