Our formula for the area of any triangle can be explained if we take a look at a “right” triangle.

Just look at the images below—a rectangle that comes with the same base and height as our original triangle.

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### Video Transcription

The area of a right triangle is one-half of the area of the rectangle! Because the areas of two (2) congruent triangles are exactly identical to our area of the rectangle, we can use the formula \(A=\frac{1}{2}\: b\cdot h\) to determine the area of the triangle.

When we’ll be using our formula for triangles when we’re looking to find the area, it is important to identify the triangle’s base and corresponding height (that is perpendicular to the triangle’s base).

**For example:**

The problem: Find the area of a triangle that has a height of \(4\) (four) inches and a base of \(10\) (ten) inches.

\(Area\, (A) =\frac{1}{2}\: b\cdot h\) Begin with our formula for a triangle’s area. \(A=\frac{1}{2}\: \cdot 10 \cdot 4\) Substitute \(10\) (ten) for the base and \(4\) (four) for the height. \(A=\frac{1}{2}\: \cdot 40\) Now, multiply. \(A = 20\)

The answer is: \(A\, (area) = 20\, in^{2}\)

Now let’s take a look at a trapezoid. If we want to discover the area of any trapezoid, we must take average lengths of the two \((2)\) parallel bases, and then multiply the average length by its height: \(A=\frac{b_1+b_2}{2}\: h\)

Look at the example provided below. Do you notice that a trapezoid’s height is always perpendicular to its bases (just as when we find a parallelogram’s height)?

**For example:**

Problem: We must find the area of this trapezoid:

\(A=\frac{b_1+b_2}{2}\: h\) We begin with the formula for a trapezoid’s area. \(\begin{align*} A&=\frac{(4+7)}{2}\cdot 2\\ \\A&=\frac{11}{2}\cdot 2\\ \\A&= 11 \\ \end{align*} \)

We substitute \(4\) (four) and \(7\) (seven) for the bases and \(2\) (two) for the height and then find \(A\).

The answer is: The area of our trapezoid is \(11\,cm^{2}\).

**Area Formulas**

We can use the following formulas if we need to find the areas of different shapes:

For a square: \(A = s^{2}\)

For a rectangle: \(A = l \cdot w\)

For a parallelogram: \(A = b \cdot h\)

For a triangle: \(A =\frac{1}{2}\: b\cdot h\)

For a trapezoid: \(A=\frac{b_1+b_2}{2}\: h\)

**Working with Perimeter & Area**

Often, we are asked to find the perimeter or area of shapes that are not standard polygons. Architects and artists, for example, generally are dealing with complex shapes. However, these complex shapes may be thought of as being the composition of less complicated, smaller shapes such as rectangles, triangles, or trapezoids.

If we want to find the perimeter of a non-standard shape, we can still determine the distance around that shape if we add together the lengths of all sides.

To find the area of a non-standard shape, we need to do it a bit differently. We need to create different regions within that shape for which we then can find the areas that we then can add together. Just take a look at how we’ve done this below.

**The example:**

Our problem is: Find the perimeter and area of this polygon.

The perimeter

(P) = 18 + 6 + 3 + 11 + 9.5 + 6 + 6

P = 59.5 cm

To find our perimeter, add together all lengths of the shape’s sides. Begin at the top to work clockwise around our shape.

To find our shape’s area, we can divide our polygon into two \((2)\) separate and simpler regions. Now, the area of the whole shape, the entire polygon, will be equal to the sum of the areas of these two regions.

Area of our polygon = (Area of region A) + (Area of region B)

The area of region \(A \)

\(Area (A) = l\cdot w\)

Region A is a rectangle.

To find our area, we multiply the length (18) by width (6).

\(A = 18\cdot 6\) \(A = 108\)Region A’s area is: \(108\, cm^{2}\).

Area of Region B

Area (A) =\(\frac{1}{2}\: b\cdot h\)

Region \(B\) is a triangle.

To find our area, we use this formula:

\(\frac{1}{2}\: bh \) \(A =\frac{1}{2}\: \cdot 9\cdot 9\)where the base is \(9\) (nine) and the height is \(9\) (nine).

\(A = \frac{l }{2}\cdot 81\) \(A = 40.5\)Region B’s area is: \( 40.5 \, cm^{2}\).

Now add both regions together:

\(108\, cm^{2} \,plus \, 40.5 \, cm^{2} \) \(= 148.5\, cm^{2} \).The answer is:

The perimeter = \(59.5 \,cm\)

The area = \(= 148.5\, cm^{2} \).

We may also use what we know about perimeters and areas to help us solve problems in situations where we are buying paint or fencing, or to determine how big a rug should be for our living room. Here’s an example of fencing.

**The example**

The problem: Rosie is planting her garden that has the dimensions as shown below. Now, she wants to put an even, thin layer of mulch across the whole surface of her garden. The price of the mulch is $3 per square foot. What’s the amount of money she’ll need to spend on the mulch?

Well, this shape combines two simpler shapes, a trapezoid, and a rectangle. First, we need to find the area of each shape.

\(A = l \cdot w\)

\(A = 8 \cdot 4\) So first, we find the rectangle’s area.

\(A = 32 ft^{2}\) \(\begin{align*}

A&=\frac{b\,_1+b\,_2}{2}\\

\\A&=\frac{14+8}{2}\cdot 4\\

\\A&=\frac{22}{2}\cdot 4 \\

\\A&= 11\cdot 4\\

\\A&=44\, ft^{2}\\

\end{align*}

\)

Then we find the trapezoid’s area.

\(32 \,ft^{2}\, plus\, 44 ft^{2} = 76\, ft^{2}\) Add the two measurements. \(76 \,ft^{2} \cdot \$3 = \$228\) We are multiplying by \(\$3\) to see how much Rosie will be spending.The answer is: Rosie will be spending \(\$228\) for covering the entire garden with mulch.

Now, find the area of this shape shown below:

A) \(11\, ft^{2}\)

B) \(18\, ft^{2}\)

C) \(20.3\, ft\)

D) \(262.8\, ft^{2}\)

**Which is the correct answer?**

A) \(11\, ft^{2}\) This correct. Our shape is a trapezoid, so we can use this formula: \(A=\frac{b_1+b_2}{2}\: h\) for finding the area: \(A=\frac{2+9}{2}\cdot 2\)

B) \(18\, ft^{2}\)
Incorrect. It seems like you’ve multiplied \(2\) by \(9\) to come up with \(18\, ft^{2}\)^{. }This would have worked if our shape would have been a rectangle but our shape is a trapezoid. So we need to use this formula: \(A=\frac{b_1+b_2}{2}\: h\). The correct answer would be \(11\, ft^{2}\).

C) \(20.3\, ft\) Incorrect. It seems like you’ve added all of the dimensions together. Well, that gives the perimeter. For finding the trapezoid’s area, we use this formula: \(A=\frac{b_1+b_2}{2}\: h\). Our correct answer would be: \(11\, ft^{2}\).

D) \(262.8\, ft^{2}\) Incorrect. It seems like you’ve multiplied all dimensions together. Our shape is a trapezoid, which means we have to use this formula: . The correct answer would be: \(11\, ft^{2}\).

**Summary**

A \(2-dimensional\, shape’s\) perimeter is the distance around that shape. We can find it by adding up all of the sides (they all have to be the same measurement units, though).

The area of a \(2-dimensional\, shape\) can be found by counting the number of all squares that are covering the shape. There are quite a few formulas for quickly finding the area of some standard polygons, like parallelograms and triangles.