Unit 1 Area And Surface Area — Unit Plan

Here are two important principles for finding area:

1. If two figures can be placed one on top of the other so that they match up exactly, then they have the same area.

2. We can decompose a figure (break a figure into pieces) and rearrange the pieces (move the pieces around) to find its area.

Here are illustrations of the two principles.

• Each square on the left can be decomposed into 2 triangles. These triangles can be rearranged into a large triangle. So, the large triangle has the same area as the 2 squares.

• Similarly, the large triangle on the right can be decomposed into 4 equal triangles. The triangles can be rearranged to form 2 squares. If each square has an area of 1 square unit, then the area of the large triangle is 2 square units. We also can say that each small triangle has an area of $\frac12$ square unit.

A parallelogram is a quadrilateral (it has four sides). The opposite sides of a parallelogram are parallel. The opposite sides of a parallelogram have the same length, and the opposite angles of a parallelogram have the same measure in degrees.

There are several strategies for finding the area of a parallelogram.

• We can decompose and rearrange a parallelogram to form a rectangle. Here are three ways:

  

  Three identical parallelograms on separate grids, each has a base of four units and a height of three units. First parallelogram, vertical dashed segment extending from the bottom left vertex to the opposite side, forming a triangle. An arrow extends from the triangle to the opposite side of the parallelogram to create a rectangle 4 units wide and 3 units high. Second parallelogram, vertical dashed segment extending from the top right vertex to the opposite side, forming a triangle. An arrow extends from the triangle to the opposite side of the parallelogram to create a rectangle 4 units wide and 3 units high. Third parallelogram, vertical dashed segment through the middle of the parallelogram. An arrow extends from the resulting shape to the opposite side of the parallelogram to create a rectangle 4 units wide and 3 units high.

• We can enclose the parallelogram and then subtract the area of the two triangles in the corner.

  

  Two drawings of parallelograms on grids. On the left, a triangle in the bottom left and top right corner is in white and the center is colored blue. On the right, the image from the left is repeated, but the triangles in the corners are colored yellow. Arrows are drawn from the triangles to the right. The triangles two triangles are joined on the right to form a rectangle.

Both of these ways will work for any parallelogram. However, for some parallelograms the process of decomposing and rearranging requires a lot more steps than if we enclose the parallelogram with a rectangle and subtract the combined area of the two triangles in the corners.

A shaded parallelogram on a grid. Base of three units. Slanted sides that decline 6 vertical units over 9 horizontal units. Parallelogram decomposed by dashed segments into six equal right triangles. Each triangle has a vertical side of 2 units and horizontal side of 3 units. Arrows extend to the left from each of the lower 5 triangles. The resulting shape is a rectangle that is 6 units tall by 3 units wide.

• We can choose any side of a parallelogram as the base. Both the side selected (the segment) and its length (the measurement) are called the base.

• If we draw any perpendicular segment from a point on the base to the opposite side of the parallelogram, that segment will always have the same length. We call that value the height. There are infinitely many segments that can represent the height!

Here are two copies of the same parallelogram.

2 copies of the same parallelogram. On the left, base = 6 units. Corresponding height = 4 units. On the right, base = 5 units. Corresponding height = 4.8 units. For both, 3 different segments are shown to represent the height.

For both, three different segments are shown to represent the height. We could draw in many more!

No matter which side is chosen as the base, the area of the parallelogram is the product of that base and its corresponding height. We can check this:

We can see why this is true by decomposing and rearranging the parallelograms into rectangles.

Notice that the side lengths of each rectangle are the base and height of the parallelogram. Even though the two rectangles have different side lengths, the products of the side lengths are equal, so they have the same area! And both rectangles have the same area as does the parallelogram.

We often use letters to stand for numbers. If $b$ is a base of a parallelogram (in units), and $h$ is the corresponding height (in units), then the area of the parallelogram (in square units) is the product of these two numbers:

$\displaystyle b \boldcdot h$

Notice that we write the multiplication symbol with a small dot instead of a $\times$ symbol. This is so that we don’t get confused about whether $\times$ means multiply, or whether the letter $x$ is standing in for a number.

Any pair of a base and a corresponding height can help us find the area of a parallelogram, but some base-height pairs are more easily found than others.

When a parallelogram is drawn on a grid and has horizontal sides, we can use a horizontal side as the base. When it has vertical sides, we can use a vertical side as the base. The grid can help us find (or estimate) the lengths of the base and of the corresponding height.

Two parallelograms drawn on two grids. First parallelogram, 2 horizontal sides each 8 units long, 2 slanted sides that rise 2 vertical units over 4 horizontal units. Bottom horizontal side labeled, b. A 2-unit perpendicular segment labeled, h, connects the horizontal sides. Second parallelogram, 2 vertical sides each 6 units long, 2 slanted sides that rise 4 vertical units over 4 horizontal units. The left vertical side is labeled, b. A 4-unit perpendicular segment labeled, h, connects one vertex of the vertical side to a point on the other vertical side.

When a parallelogram is not drawn on a grid, we can still find its area if we know a base and a corresponding height.

In this parallelogram, the corresponding height for the side that is 10 units long is not given, but the height for the side that is 8 units long is given. This base-height pair can help us find the area.

Parallelograms that have the same base and the same height will have the same area; the product of the base and height will be equal. Here are 4 different parallelograms with the same pair of base-height measurements.

A parallelogram can always be decomposed into two identical triangles by a segment that connects opposite vertices.

Going the other way around, two identical copies of a triangle can always be arranged to form a parallelogram, regardless of the type of triangle being used. To produce a parallelogram, we can join a triangle and its copy along any of the three sides that match, so the same pair of triangles can make different parallelograms. Here are examples of how two copies of both Triangle A and Triangle F can be composed into three different parallelograms.

This special relationship between triangles and parallelograms can help us reason about the area of any triangle.

We can reason about the area of a triangle by using what we know about parallelograms. Here are three general ways to do this:

• Make a copy of the triangle and join the original and the copy along an edge to create a parallelogram. Because the two triangles have the same area, one copy of the triangle has half the area of that parallelogram.

  

  Two figures labeled A, and B. Figure A is a triangle. Figure B is the same triangle as figure A with a copy along the edge of the original to create a rectangle. The right side of the rectangle is labeled 2 units, and the bottom is labeled 8 units. 

  The area of Parallelogram B is 16 square units because the base is 8 units and the height 2 units. The area of Triangle A is half of that, which is 8 square units.

  

  Two figures labeled C, and D. Figure C is another triangle, and figure D is the same triangle as figure C, with a copy along the edge of the original to create parallelogram. The left base of the parallelogram is labeled 4 units, and the height is labeled 6 units.

  The area of Parallelogram D is 24 square units because the base is 4 units and the height 6 units. The area of Triangle C is half of that, which is 12 square units.

• Decompose the triangle into smaller pieces and compose them into a parallelogram.

  

  Two images of a triangle. Image on right has a dashed line cutting off the top portion. Image on left has the cut off portion moved next to the bottom of the triangle to create a parallelogram. An arrow indicating that the cut off portion from other image was moved.

  In the new parallelogram, $b = 6$, $h = 2$, and $6 \boldcdot 2 = 12$, so its area is 12 square units. Because the original triangle and the parallelogram are composed of the same parts, the area of the original triangle is also 12 square units.

• Draw a rectangle around the triangle. Sometimes the triangle has half of the area of the rectangle.

  

  The large rectangle can be decomposed into smaller rectangles. Each smaller rectangle can be decomposed into two right triangles.

  • The rectangle on the left has an area of $4 \boldcdot 3$, or 12, square units. Each right triangle inside it is 6 square units in area.

  • The rectangle on the right has an area of $2 \boldcdot 3$, or 6, square units. Each right triangle inside it is 3 square units in area.

  • The area of the original triangle is the sum of the areas of a large right triangle and a small right triangle: 9 square units.

  Sometimes, the triangle is half of what is left of the rectangle after removing two copies of the smaller right triangles.

  

  Three images of the same triangle. The first image is the triangle alone. The second is the triangle surrounded by a rectangle. The third image is of the triangle now with a copy composed into a parallelogram within the rectangle, with arrows drawing the remaining parts of the rectangle into a smaller rectangle.

  The right triangles being removed can be composed into a small rectangle with area $(2 \boldcdot 3)$ square units. What is left is a parallelogram with area $5 \boldcdot 3 - 2 \boldcdot 3$, which equals $15-6$, or 9, square units.

  Notice that we can compose the same parallelogram with two copies of the original triangle! The original triangle is half of the parallelogram, so its area is $\frac12 \boldcdot 9$, or 4.5, square units.

• We can choose any of the three sides of a triangle to call the base. The term “base” refers to both the side and its length (the measurement).
• The corresponding height is the length of a perpendicular segment from the base to the vertex opposite it. The opposite vertex is the vertex that is not an endpoint of the base.

Here are three pairs of bases and heights for the same triangle. The dashed segments in the diagrams represent heights.

A segment showing a height must be drawn at a right angle to the base, but it can be drawn in more than one place. It does not have to go through the opposite vertex, as long as it connects the base and a line that is parallel to the base and goes through the opposite vertex, as shown here.

The base-height pairs in a triangle are closely related to those in a parallelogram. Recall that two copies of a triangle can be composed into one or more parallelograms. Each parallelogram composed of the triangle and its copy shares at least one base with the triangle.

Two identical triangles, each with a copy composing the triangle into two different parallelograms. In each parallelogram has the bottom side labeled “base” and dashed lines at right angles to the base indicating the height of the parallelogram.

For any base that they share, the corresponding height is also shared, as shown by the dashed segments.

We can use the base-height measurements and our knowledge of parallelograms to find the area of any triangle.

• The formula for the area of a parallelogram with base $b$ and height $h$ is $b \boldcdot h$.
• A triangle takes up half of the area of a parallelogram with the same base and height. We can therefore express the area, $A$, of a triangle as: $\displaystyle A = \frac12 \boldcdot b \boldcdot h$

  

  

• The area of Triangle A is 15 square units because $\frac12 \boldcdot 5 \boldcdot 6=15$.

• The area of Triangle B is 4.5 square units because $\frac12 \boldcdot 3 \boldcdot 3 = 4.5$. 

• The area of Triangle C is 24 square units because $\frac12 \boldcdot 12 \boldcdot 4 = 24$.

In each case, one side of the triangle is the base but neither of the other sides is the height. This is because the angle between them is not a right angle.  

In right triangles, however, the two sides that are perpendicular can be a base and a height.

The area of this triangle is 18 square units whether we use 4 units or 9 units for the base.

• The surface area of a figure (in square units) is the number of unit squares it takes to cover the entire surface without gaps or overlaps.
• If a three-dimensional figure has flat sides, the sides are called faces.
• The surface area is the total of the areas of the faces.

For example, a rectangular prism has six faces. The surface area of the prism is the total of the areas of the six rectangular faces.

So the surface area of a rectangular prism that has edge-lengths of 2 cm, 3 cm, and 4 cm has a surface area of

$\displaystyle (2\boldcdot 3)+ (2\boldcdot 3) + (2\boldcdot 4) + (2\boldcdot 4) + (3\boldcdot 4) + (3\boldcdot 4)$

or 52 square centimeters.

A net of a pyramid has one polygon that is the base. The rest of the polygons are triangles. A pentagonal pyramid and its net are shown here.

A net of a prism has two copies of the polygon that is the base. The rest of the polygons are rectangles. A pentagonal prism and its net are shown here.

In a rectangular prism, there are three pairs of parallel and identical rectangles. Any pair of these identical rectangles can be the bases.

Because a net shows all the faces of a polyhedron, we can use it to find its surface area. For instance, the net of a rectangular prism shows three pairs of rectangles: 4 units by 2 units, 3 units by 2 units, and 4 units by 3 units.

The surface area of the rectangular prism is 52 square units because $8+8+6+6+12+12=52$.