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Area represents the amount of space enclosed within a two-dimensional figure. The method of calculating the area of a figure depends on the type of figure. This lesson will show how the formulas for the area of some figures can be derived from the formula for the area of a rectangle. Buckle up — the journey is about to begin!

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Rectangle
Triangle
Altitude of a Triangle
Polygon
Coordinate Plane
Distance Formula
Two-dimensional Figure

Challenge

Dividing a Rectangle in Half

Consider a rectangle with length

l

and width

w .

The area of the rectangle is the product of its dimensions,

A = l w .

If a diagonal is drawn across the rectangle, the shape is divided into two triangles.

Write a formula for finding the area of the bottom triangle.

Discussion

Deriving the Formula for the Area of a Triangle

A diagonal of a rectangle divides the rectangle into two right triangles. Because of this, a formula for the area of a right triangle can be derived from the formula for the area of a rectangle. The good news is that the same formula applies to any type of triangle!

Rule

Area of a Triangle

The area of a triangle is half the product of its base $b$ and its height $h .$

$A = \frac{1}{2} b h$

The triangle's base can be any of its sides. The height – or altitude – of the triangle is the segment that is perpendicular to the base and connects the base or its extension with its opposite vertex.

Proof

Proof for Right Triangles

First, consider the particular case of a right triangle. It is always possible to reflect a right triangle across its hypotenuse to form a rectangle.

Note that the area of the rectangle formed is twice the area of the original right triangle. Because of this, the formula for the area of the rectangle,

A_{r} = ℓ w,

can be used to find the area of the right triangle.

A_{r} = 2 A_{t} \Rightarrow ℓ w = 2 A_{t}

Furthermore, the height and base of the right triangle have the same measures as the width and length of the rectangle formed by reflecting the triangle. Based on this observation,

b

and

h

can be substituted for

ℓ

and

w,

respectively, to solve for the area of the original right triangle in terms of its base and height.

ℓ w = 2 A_{t}

SubstituteII

$ℓ = b$ , $w = h$

b h = 2 A_{t}

▼

Solve for

A_{t}

Div

Divide by $2$

\frac{b h}{2} = A_{t}

MoveNumRight

$\frac{a}{b} = \frac{1}{b} \cdot a$

\frac{1}{2} b h = A_{t}

RearrangeEqn

Rearrange equation

A_{t} = \frac{1}{2} b h

This shows that the area of a right triangle can be calculated by using the formula

A = \frac{1}{2} b h .

Proof for Non-Right Triangles

To generalize the previous result, it is useful to note that any non-right triangle can be split into two right triangles by drawing one of its heights.

Note that the area of the non-right triangle

A

is equal to the sum of the individual areas of the smaller right triangles

A_{1}

and

A_{2} .

Therefore, it is possible to calculate the area of the non-right triangle by using the previous result for the areas of the smaller right triangles.

A = \frac{b _{1} h}{2} + \frac{b _{2} h}{2}

▼

Simplify right-hand side

AddFrac

Add fractions

A = \frac{b _{1} h + b _{2} h}{2}

FactorOut

Factor out $h$

A = \frac{( b _{1} + b _{2} ) h}{2}

MoveNumRight

$\frac{a}{b} = \frac{1}{b} \cdot a$

A = \frac{1}{2} (b_{1} + b_{2}) h

Substitute

$b_{1} + b_{2} = b$

A = \frac{1}{2} b h

It has been found that the area of the non-right triangle is half the product of its base

b

and its height

h .

This is the same result as the area for a right triangle. Therefore, the area of any triangle is half the product of its base

b

and its height

h .

$A = \frac{1}{2} b h$

Example

Tangram Puzzle

On his birthday, Mark's uncle gave him a tangram, a Chinese puzzle made of seven polygons that can be used to create different shapes. The seven individual pieces are called tans.

Mark's uncle warned him that once the pieces are taken out of the box, putting them back is a challenge.

a After trying for a while, Mark managed to form a cat.

Find the area of the cat's hind leg, Tan

1 .

b Mark's uncle showed him how to form a running person.

If the right foot has an area of

4

square centimeters, what is the value of

x ?

Hint

a Piece

1

has the shape of a triangle, so use the formula for the area of a triangle.

b Use the formula for the area of a triangle and solve it for the height.

Solution

a The cat's hind leg has the shape of a triangle. The area of a triangle is half the base times the height.

A_{△} = \frac{1}{2} b h

The triangle's base can be any of its sides. The height is the segment that is perpendicular to the base and connects the base with its opposite vertex.

From the diagram, the base of the hind leg is

8

centimeters and the height is

4

centimeters. Substitute these values into the formula to find the area of Tan

1 .

A_{△} = \frac{1}{2} b h

SubstituteII

$b = 8$ , $h = 4$

A_{△} = \frac{1}{2} \cdot 8 \cdot 4

Multiply

A_{△} = \frac{1}{2} \cdot 32

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

A_{△} = \frac{1 \cdot 3 2}{2}

IdPropMult

Identity Property of Multiplication

A_{△} = \frac{3 2}{2}

CalcQuot

Calculate quotient

A_{△} = 16

The area of the cat's hind leg is

16

square centimeters.

b The right foot of the running person is another triangle, so begin by recalling the formula for the area of a triangle again.

A_{△} = \frac{1}{2} b h

The

4 -

centimeter-long side acts as a base of the triangle, so the

x -

centimeter-long segment is its corresponding height.

Therefore, substitute

4

for the area,

4

for the base, and

x

for the height into the formula. Then, solve the equation for

x

to find the height of the triangle.

A_{△} = \frac{1}{2} b h

SubstituteValues

Substitute values

4 = \frac{1}{2} \cdot 4 \cdot x

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

4 = \frac{4}{2} \cdot x

CalcQuot

Calculate quotient

4 = 2 x

DivEqn

$LHS / 2 = RHS / 2$

2 = x

RearrangeEqn

Rearrange equation

x = 2

The height of the triangle is

2

centimeters.

Pop Quiz

Finding the Missing Dimension

For the given triangle, find the missing dimension. Round the answer to two decimal places if necessary.

Discussion

Parallelograms

A parallelogram is a quadrilateral with two pairs of parallel sides. Parallelograms can be divided into three main types: rectangles, rhombuses, and squares.

The following properties are true for all parallelograms.

Property	Justification
The opposite sides are congruent	Parallelogram Opposite Sides Theorem
The opposite angles are congruent	Parallelogram Opposite Angles Theorem
The diagonals bisect each other	Parallelogram Diagonals Theorem

These properties are illustrated graphically in the next diagram.

Discussion

Area of a Parallelogram

The area of a parallelogram is equal to the product of its base $b$ and height $h .$ The base can be any side of the parallelogram and the height is the perpendicular distance to the opposite side.

Example

Tangram Animals

Mark continued playing with the tangram and learned to make different animal shapes, including a swan and a rabbit.

a Find the area of the swan's neck, made up of Tan

7 .

b The rabbit's body is made of Tans

1

and

2

and has an area of

32

square centimeters. Find

w .

Hint

a The swan's neck is a parallelogram.

b The rabbit's body is a parallelogram.

Solution

a The swan's neck is represented by Tan

7,

which is a parallelogram.

The area of a parallelogram is the product of its base and height. The base can be any side and the height is the perpendicular distance to the opposite side.

A = b h

For Tan

7,

the longest sides measure

4

centimeters and the perpendicular distance between them is

2

centimeters. Therefore, to find the area of the piece, substitute

4

for

b

and

2

for

h

into the formula and simplify.

A = 4 \cdot 2 \Rightarrow A = 8

The area of the swan's neck is

8

square centimeters.

b The way Tans

1

and

2

are placed forms a large parallelogram.

The longest sides have a length of

8

centimeters, and the shape has an area of

32

square centimeters. Substitute these values into the formula for the area of a parallelogram to determine the height, represented by

w .

A = b h

SubstituteValues

Substitute values

32 = 8 w

DivEqn

$LHS / 8 = RHS / 8$

4 = w

RearrangeEqn

Rearrange equation

w = 4

The height of the large parallelogram is

4

centimeters.

Discussion

Parallelograms With Congruent Sides

Parallelograms can be divided into three main types: rectangles, rhombuses, and squares. It is the time to learn about rhombuses.

Concept

Rhombus

A rhombus is a parallelogram with four congruent sides. In other words, a rhombus is a quadrilateral with two pairs of parallel sides, all four of which have the same length.

Rhombuses have some special properties that not all the parallelograms have. For instance, the diagonals of a rhombus bisect each other at a right angle. They also bisect the opposite angles.

Rhombuses are symmetric about both diagonals.

A rhombus with four right angles is called a square.

Discussion

Area of a Rhombus

The area of a rhombus is half the product of the lengths of the diagonals.

Alternatively, since a rhombus is a parallelogram, its area can also be calculated by multiplying its base and height.

Example

Rhombuses Everywhere

Since Mark received the tangram puzzle, he sees polygons everywhere.

a Mark went to a baseball game with his family last Sunday. At one point, the game was stopped because second base was not in the right spot. Mark then realized that the bases form a rhombus.

Home plate and second base are about

127

feet apart, as are first and third bases. What is the area of the rhombus formed by the bases? Round the answer to the nearest hundred.

b While walking to his parent's car in the parking lot, Mark saw a car's logo that was made of three identical rhombuses.

The diagonals of each rhombus are about

5

and

8.7

centimeters long. If the perpendicular distance between opposite sides is

4.3

centimeters, what is the side length of each rhombus? Round the answer to one decimal place.

Hint

a The area of a rhombus the half the product of its diagonals.

b Use the lengths of the diagonals to find the area of each rhombus. Then, use the formula for the area of a parallelogram to determine the length of the base.

Solution

a The bases form a rhombus whose area can be found by using the following formula.

A = \frac{1}{2} d_{1} d_{2}

The diagonals of the infield are formed by connecting the home plate and the second base, and also the first and third base.

The length of each diagonal is about

127

feet. Substitute this value into the formula for the area.

A = \frac{1}{2} d_{1} d_{2}

SubstituteII

$d_{1} = 127$ , $d_{2} = 127$

A = \frac{1}{2} \cdot 127 \cdot 127

Multiply

A = \frac{1}{2} \cdot 16129

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

A = \frac{1 6 1 2 9}{2}

CalcQuot

Calculate quotient

A = 8064.5

The area of the rhombus formed by the bases, rounded to the nearest hundred, is

8100

square feet.

b Let

s

be the rhombus side length. Notice that the perpendicular distance between opposite sides is the height of the rhombus. Since a rhombus is a parallelogram, its area can be found by multiplying its base and height.

A = b h \Rightarrow A = s \cdot 4.3

The side length can be found by first determining the area of each rhombus. The area of a rhombus is also equal to half the product of the diagonals.

A = \frac{1}{2} d_{1} d_{2}

It is given that the diagonals are

5

and

8.7

centimeters long. The area of each rhombus can be found by substituting these values into the previous formula.

A = \frac{1}{2} d_{1} d_{2}

SubstituteII

$d_{1} = 8.7$ , $d_{2} = 5$

A = \frac{1}{2} \cdot 8.7 \cdot 5

Multiply

A = \frac{1}{2} \cdot 43.5

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

A = \frac{4 3 . 5}{2}

CalcQuot

Calculate quotient

A = 21.75

Finally, substitute the area into the equation written at the beginning and solve it for

s .

A = s \cdot 4.3

Substitute

$A = 21.75$

21.75 = s \cdot 4.3

DivEqn

$LHS / 4.3 = RHS / 4.3$

\frac{2 1 . 7 5}{4 . 3} = s

CalcQuot

Calculate quotient

5.058139 \dots = s

RearrangeEqn

Rearrange equation

s = 5.058139 \dots

RoundDec

Round to $1$ decimal place(s)

s \approx 5.1

The side length of each rhombus is about

5.1

centimeters.

Pop Quiz

Solving Parallelograms and Rhombuses

The following applet shows a general parallelogram or rhombus. Calculate the missing dimension of the given polygon. Round the answer to two decimal places.

Discussion

Trapezoids

A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called the bases of the trapezoid, and the two other sides are called the legs. Two angles that have a base as a common side are called the base angles.

Trapezoids with congruent legs have a special name.

Concept

Isosceles Trapezoid

An isosceles trapezoid is a trapezoid whose legs are congruent.

Isosceles trapezoids have two main properties.

Property	Justification
The diagonals are congruent.	Isosceles Trapezoid Diagonals Theorem
Each pair of base angles is congruent.	Isosceles Trapezoid Base Angles Theorem

Discussion

Area of a Trapezoid

The area of a trapezoid is half the height times the sum of the lengths of the bases. In other words, the area of a trapezoid is the height multiplied by the average of the bases.

$A = \frac{1}{2} h (b_{1} + b_{2})$

Extra

Graphical Derivation

The formula for the area of a trapezoid with bases

b_{1}

and

b_{2}

and height

h

can be derived by transforming the trapezoid into a triangle with base

b_{1} + b_{2}

and height

h .

Example

A Head Full of Polygons

Mark is getting ready to go to school. As he eats breakfast with his parents, he looks up and begins to see trapezoids everywhere.

a The lampshade has bases that are

12

and

6

inches long and it is

10

inches tall.

What is the area of the lampshade?

b When Mark saw Donatello, his pet turtle, he got the idea to make a turtle out of his tangram.

The bases of the shell are about

4.7

and

3.1

inches long and it has an area of

6.2

square inches. How tall is the turtle's shell? Round the answer to one decimal place.

Hint

a The lampshade is a trapezoid.

b The area of a trapezoid is one-half the product of the height and the sum of the bases. Solve this formula for the height.

Solution

a The shade of the lamp is a trapezoid. Therefore, its area is half the height times the sum of the length of the bases.

A = \frac{1}{2} h (b_{1} + b_{2})

It is said that the bases o

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Catch-Up and Review

Proof

Proof for Right Triangles

Proof for Non-Right Triangles

Hint

Solution

Hint

Solution

Hint

Solution

Extra

Hint

Solution