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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.

Challenge

Dividing a Rectangle in Half

Consider a rectangle with length and width The area of the rectangle is the product of its dimensions, If a diagonal is drawn across the rectangle, the shape is divided into two triangles.
Rectangle with a diagonal
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 and its height

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.

Triangles

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.
Showing right triangle as half 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, can be used to find the area of the right triangle.
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, and can be substituted for and respectively, to solve for the area of the original right triangle in terms of its base and height.
Solve for
This shows that the area of a right triangle can be calculated by using the formula

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.
Scalene triangle split into two right triangles
Note that the area of the non-right triangle is equal to the sum of the individual areas of the smaller right triangles and 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.
Simplify right-hand side
It has been found that the area of the non-right triangle is half the product of its base and its height 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 and its height

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.

Tangram

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.
Tangram - Cat
Find the area of the cat's hind leg, Tan
b Mark's uncle showed him how to form a running person.
Tangram - Person running
If the right foot has an area of square centimeters, what is the value of

Hint

a Piece 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.
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.
Tangram - Cat
From the diagram, the base of the hind leg is centimeters and the height is centimeters. Substitute these values into the formula to find the area of Tan
The area of the cat's hind leg is 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.
The centimeter-long side acts as a base of the triangle, so the centimeter-long segment is its corresponding height.
Tangram - Person running
Therefore, substitute for the area, for the base, and for the height into the formula. Then, solve the equation for to find the height of the triangle.
The height of the triangle is centimeters.
Pop Quiz

Finding the Missing Dimension

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

Random triangles.
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.
Parallelogram
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.

Parallelogram
Discussion

Area of a Parallelogram

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

Parallelogram, A=bh
Example

Tangram Animals

Mark continued playing with the tangram and learned to make different animal shapes, including a swan and a rabbit.
Tangram - swan and rabbit
a Find the area of the swan's neck, made up of Tan
b The rabbit's body is made of Tans and and has an area of square centimeters. Find

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 which is a parallelogram.
Tangram - swan
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.
For Tan the longest sides measure centimeters and the perpendicular distance between them is centimeters. Therefore, to find the area of the piece, substitute for and for into the formula and simplify.
The area of the swan's neck is square centimeters.
b The way Tans and are placed forms a large parallelogram.
Tangram - bunny
The longest sides have a length of centimeters, and the shape has an area of square centimeters. Substitute these values into the formula for the area of a parallelogram to determine the height, represented by
The height of the large parallelogram is 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.
Rhombus
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.
Rhombus and diagonals
Rhombuses are symmetric about both diagonals.
Rhombus and 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.

Rhombus and diagonals

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

Rhombus
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.
Baseball field
Home plate and second base are about 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.
Mitsubishi Logo
The diagonals of each rhombus are about and centimeters long. If the perpendicular distance between opposite sides is 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.
The diagonals of the infield are formed by connecting the home plate and the second base, and also the first and third base.
Baseball field
The length of each diagonal is about feet. Substitute this value into the formula for the area.
The area of the rhombus formed by the bases, rounded to the nearest hundred, is square feet.
b Let 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.
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.
It is given that the diagonals are and centimeters long. The area of each rhombus can be found by substituting these values into the previous formula.
Finally, substitute the area into the equation written at the beginning and solve it for
The side length of each rhombus is about 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.

Random parallelograms and rhombus.
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.

Trapezoid

Trapezoids with congruent legs have a special name.

Concept

Isosceles Trapezoid

An isosceles trapezoid is a trapezoid whose legs are congruent.

An isosceles trapezoid is depicted with parallel upper and lower sides, and equal-length sides on the right and left.

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.

Extra

Graphical Derivation
The formula for the area of a trapezoid with bases and and height can be derived by transforming the trapezoid into a triangle with base and height
Converting a trapezoid into a triangle
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.

Objects in a house
a The lampshade has bases that are and inches long and it is inches tall.
Lamp
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.
Tangram - Turtle
The bases of the shell are about and inches long and it has an area of 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.