9. Areas of Parallelograms, Triangles, and Trapezoids
Sign In
An error ocurred, try again later!
Chapter 9
9.
Areas of Parallelograms, Triangles, and Trapezoids
This lesson explores methods for calculating the area of various geometric shapes, including parallelograms, triangles, trapezoids, and rhombuses. It provides insights into their properties and offers practical guidance to help learners solve geometry problems effectively.
Show less Show more expand_more 
Lesson Settings & Tools
| | 18 Theory slides |
| | 8 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Areas of Parallelograms, Triangles, and Trapezoids
Slide of 18
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
Share
Report error
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=lw. 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.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["l","w"],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"A_(\u25b3)=","formTextAfter":null,"answer":{"text":["\\dfrac{1}{2}lw","\\dfrac{1}{2}wl","0.5lw"]}}
Discussion
Share
Report error
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
Share
Report error
Area of a Triangle
The area of a triangle is half the product of its base b and its height h.
A=1/2bh
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.
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.
A = b_1h/2 + b_2h/2
▼
Simplify right-hand side
AddFrac
Add fractions
A = b_1h + b_2h/2
FactorOut
Factor out h
A = (b_1 + b_2)h/2
MoveNumRight
a/b=1/b* a
A = 1/2(b_1+b_2)h
Substitute
b_1+b_2= b
A = 1/2 bh
A = 1/2bh
Example
Share
Report error
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.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"A=","formTextAfter":"cm^2","answer":{"text":["16"]}}
b Mark's uncle showed him how to form a running person.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"x=","formTextAfter":"cm","answer":{"text":["2"]}}
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_(△) = 1/2bh 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.
A_(△) = 1/2bh
SubstituteII
b= 8, h= 4
A_(△) = 1/2* 8* 4
Multiply
Multiply
A_(△) = 1/2* 32
MoveRightFacToNum
a/c* b = a* b/c
A_(△) = 1* 32/2
IdPropMult
Identity Property of Multiplication
A_(△) = 32/2
CalcQuot
Calculate quotient
A_(△) = 16
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_(△) = 1/2bh The 4-centimeter-long side acts as a base of the triangle, so the x-centimeter-long segment is its corresponding height.
A_(△) = 1/2bh
SubstituteValues
Substitute values
4 = 1/2* 4* x
MoveRightFacToNum
a/c* b = a* b/c
4 = 4/2* x
CalcQuot
Calculate quotient
4 = 2x
DivEqn
.LHS /2.=.RHS /2.
2 = x
RearrangeEqn
Rearrange equation
x = 2
Pop Quiz
Share
Report error
Finding the Missing Dimension
Discussion
Share
Report error
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
Share
Report error
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
Share
Report error
Tangram Animals
Mark continued playing with the tangram and learned to make different animal shapes, including a swan and a rabbit.
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.
The height of the large parallelogram is 4 centimeters.
a Find the area of the swan's neck, made up of Tan 7.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"A=","formTextAfter":"cm^2","answer":{"text":["8"]}}
b The rabbit's body is made of Tans 1 and 2 and has an area of 32 square centimeters. Find w.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"w=","formTextAfter":"cm","answer":{"text":["4"]}}
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 = bh 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* 2 ⇒ 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.
A = bh
SubstituteValues
Substitute values
32 = 8 w
DivEqn
.LHS /8.=.RHS /8.
4 = w
RearrangeEqn
Rearrange equation
w = 4
Discussion
Share
Report error
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
Share
Report error
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
Share
Report error
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
Share
Report error
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.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"A\u2248","formTextAfter":"ft^2","answer":{"text":["8100"]}}
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.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"s\u2248","formTextAfter":"cm","answer":{"text":["5.1"]}}
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 = 1/2d_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.
A = 1/2d_1 d_2
SubstituteII
d_1= 127, d_2= 127
A = 1/2* 127* 127
Multiply
Multiply
A = 1/2* 16 129
MoveRightFacToNum
a/c* b = a* b/c
A = 16 129/2
CalcQuot
Calculate quotient
A = 8064.5
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 = 1/2d_1 d_2
SubstituteII
d_1= 8.7, d_2= 5
A = 1/2* 8.7 * 5
Multiply
Multiply
A = 1/2* 43.5
MoveRightFacToNum
a/c* b = a* b/c
A = 43.5/2
CalcQuot
Calculate quotient
A = 21.75
A = s* 4.3
Substitute
A= 21.75
21.75 = s* 4.3
DivEqn
.LHS /4.3.=.RHS /4.3.
21.75/4.3 = s
CalcQuot
Calculate quotient
5.058139... = s
RearrangeEqn
Rearrange equation
s = 5.058139...
RoundDec
Round to 1 decimal place(s)
s ≈ 5.1
Pop Quiz
Share
Report error
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
Share
Report error
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
Share
Report error
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
Share
Report error
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=1/2h(b_1+b_2)
Extra
Graphical DerivationThe 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
Share
Report error
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.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"A=","formTextAfter":"in^2","answer":{"text":["90"]}}
b When Mark saw Donatello, his pet turtle, he got the idea to make a turtle out of his tangram.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"h\u2248","formTextAfter":"in","answer":{"text":["1.6"]}}
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 = 1/2h(b_1+b_2)
SubstituteValues
Substitute values
A = 1/2* 10( 12+ 6)
▼
Simplify right-hand side
AddTerms
Add terms
A = 1/2* 10* 18
Multiply
Multiply
A = 1/2* 180
MoveRightFacToNum
a/c* b = a* b/c
A = 180/2
CalcQuot
Calculate quotient
A = 90
b The bases and the area of the trapezoid that makes up the turtle's shell are given.
A = 1/2h(b_1+b_2)
SubstituteValues
Substitute values
6.2 = 1/2* h( 4.7+ 3.1)
▼
Solve for h
AddTerms
Add terms
6.2 = 1/2* h* 7.8
CommutativePropMult
Commutative Property of Multiplication
6.2 = 1/2* 7.8 * h
MoveRightFacToNum
a/c* b = a* b/c
6.2 = 7.8/2 * h
MultEqn
LHS * 2/7.8=RHS* 2/7.8
6.2 * 2/7.8= h
▼
Simplify
MoveLeftFacToNum
a*b/c= a* b/c
6.2* 2/7.8= h
Multiply
Multiply
12.4/7.8= h
CalcQuot
Calculate quotient
1.589743... = h
RearrangeEqn
Rearrange equation
h = 1.589743...
RoundDec
Round to 1 decimal place(s)
h≈ 1.6
Pop Quiz
Share
Report error
Solving Trapezoids
Example
Share
Report error
Futsal Formations
Mark's futsal team has their final championship game tonight. They have been practicing different strategies for this game. The coach prepared some plays on a whiteboard with a coordinate system. One unit on the board represents 2 meters on the actual court.
a When Mark's team is defending, they arrange themselves to form a parallelogram, where the player closest to the ball will try to recover it while the rest of the team covers the spaces where the opponent could attack.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"A\u2248","formTextAfter":"m^2","answer":{"text":["92"]}}
b When Mark's team is attacking and the opponent closes the central part of the court, they attack the flanks using a trapezoidal formation.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"h\u2248","formTextAfter":"m","answer":{"text":["21.2"]}}
Hint
a Use the distance formula to find the lengths of the diagonals, then substitute them into the formula for the area of a rhombus. Remember to convert each distance to meters.
b Use the distance formula to find the lengths of the bases. Then, use the formula for the area of a trapezoid to determine the height of the trapezoid. Remember to convert each distance to meters.
Solution
a The area of a rhombus is half the product of its diagonals.
d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)
SubstitutePoints
Substitute ( -6,1) & ( 2,4)
d_1 = sqrt(( 2-( -6))^2+( 4- 1)^2)
▼
Evaluate right-hand side
SubNeg
a-(- b)=a+b
d_1 = sqrt((2+6)^2+(4-1)^2)
AddSubTerms
Add and subtract terms
d_1 = sqrt(8^2+3^2)
CalcPow
Calculate power
d_1 = sqrt(64+9)
AddTerms
Add terms
d_1 = sqrt(73)
d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)
SubstitutePoints
Substitute ( -3,5) & ( -1,0)
d_2 = sqrt(( -1-( -3))^2+( 0- 5)^2)
▼
Evaluate right-hand side
SubNeg
a-(- b)=a+b
d_2 = sqrt((-1+3)^2+(0-5)^2)
AddSubTerms
Add and subtract terms
d_2 = sqrt(2^2+(-5)^2)
CalcPow
Calculate power
d_2 = sqrt(4+25)
AddTerms
Add terms
d_2 = sqrt(29)
| Length on the Whiteboard | Length on the Court |
|---|---|
| d_1 = sqrt(73) | d_1 = 2sqrt(73) |
| d_2 = sqrt(29) | d_2 = 2sqrt(29) |
A = 1/2d_1d_2
SubstituteII
d_1= 2sqrt(73), d_2= 2sqrt(29)
A = 1/2* 2sqrt(73)* 2sqrt(29)
▼
Simplify right-hand side
CommutativePropMult
Commutative Property of Multiplication
A = 1/2* 2* 2 * sqrt(73)*sqrt(29)
Multiply
Multiply
A = 1/2* 4 * sqrt(73)*sqrt(29)
MoveRightFacToNum
a/c* b = a* b/c
A = 4/2* sqrt(73)*sqrt(29)
CalcQuot
Calculate quotient
A = 2sqrt(73)*sqrt(29)
ProdSqrt
sqrt(a)*sqrt(b)=sqrt(a* b)
A = 2sqrt(73* 29)
Multiply
Multiply
A = 2sqrt(2117)
UseCalc
Use a calculator
A = 2* 46.010868...
Multiply
Multiply
A = 92.021736...
RoundInt
Round to nearest integer
A ≈ 92
b The area of a trapezoid is half the height times the sum of the lengths of the bases.
d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)
SubstitutePoints
Substitute ( 4,5) & ( 6,-5)
b_1 = sqrt(( 6- 4)^2 + ( -5- 5)^2)
b_1 ≈ 10.2
d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)
SubstitutePoints
Substitute ( -6,1) & ( -5,-4)
b_2 = sqrt(( -5-( -6))^2 + ( -4- 1)^2)
b_2 ≈ 5.1
| Length on the Whiteboard | Length on the Court |
|---|---|
| b_1 = 10.2 | b_1 = 20.4 |
| b_2 = 5.1 | b_2 = 10.2 |
A = 1/2h(b_1+b_2)
SubstituteValues
Substitute values
324.4 = 1/2* h( 20.4+ 10.2)
▼
Solve for h
AddTerms
Add terms
324.4 = 1/2* h* 30.6
CommutativePropMult
Commutative Property of Multiplication
324.4 = 1/2* 30.6 * h
MoveRightFacToNum
a/c* b = a* b/c
324.4 = 30.6/2h
CalcQuot
Calculate quotient
324.4 = 15.3h
DivEqn
.LHS /15.3.=.RHS /15.3.
324.4/15.3 = h
RearrangeEqn
Rearrange equation
h = 324.4/15.3
CalcQuot
Calculate quotient
h = 21.202614...
RoundDec
Round to 1 decimal place(s)
h ≈ 21.2
Closure
Share
Report error
Composite Figures
Sometimes a plane figure can be made up of two or more geometric shapes. These figure are called composite figures. The area of a composite figure is the sum of the areas of all the basic figures that make it up. A good example of composite figures are those formed by a tangram puzzle. For instance, consider the following rocket.
The area of the rocket is equal to the sum of the areas of all seven polygons. However, some pieces can be considered together to perform fewer computations. For example, the rocket can be seen as three trapezoids.
The area of the rocket is the sum of the areas of the three trapezoids that make it up. A_(Rocket) = A_(7,3) + A_(1,2,4) + A_(5,6) The three areas can all be found with the same formula. All the computations are summarized in the table below.
| Tans Forming the Trapezoid | Dimensions | A=1/2h(b_1+b_2) | Area (cm^2) |
|---|---|---|---|
| 7 and 3 | b_1 = 8 b_2=4 h=2 |
A_(7,3) = 1/2* 2(8+4) | 12 |
| 1, 2, and 4 | b_1 = 12 b_2=8 h=4 |
A_(1,2,4) = 1/2* 4(12+8) | 40 |
| 5 and 6 | b_1 = 2sqrt(8) b_2=sqrt(8) h=sqrt(8) |
A_(5,6) = 1/2* sqrt(8)(2sqrt(8)+sqrt(8)) | 12 |
The area of the rocket is 12+40+12=64 square centimeters. In fact, any figure that is formed using the seven tans will also have an area of 64 square centimeters! The following applet shows more examples and allows playing with the tans.
Areas of Parallelograms, Triangles, and Trapezoids
Exercises
Consider the following diagrams and find the required information.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"A=","formTextAfter":null,"answer":{"text":["860"]}}
The area of △ PQR is 1408.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"RS=","formTextAfter":null,"answer":{"text":["44"]}}
The area of △ JKL is 650.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"P=","formTextAfter":null,"answer":{"text":["124"]}}
security
report
code
security
report
code
The area of a triangle is half the product of its base and height. A = 1/2bh The base can be any side of the triangle and the height is the segment that is perpendicular to the base and connects the base or its extension with its opposite vertex. In the given diagram, we can see that the three heights of the triangle are given.
When finding the area, it is important to use the correct data pair since the height is different for each base. Let's pair each base to its corresponding height.
| Base | Height |
|---|---|
| 50 | 34.4 |
| 80 | 21.5 |
| 40 | 43 |
We can find the area of △ ABC by using any of these three pairs of values. For simplicity, let's use the data from the third row.
The area of the triangle is 860 square units.
Let's start by noticing that RS is perpendicular to PQ. If we take PQ as the base of the triangle, then RS is the corresponding height.
The area of a triangle is half the product of its base and height, so let's write an equation in terms of RS. A = 1/2bh ⇒ A = 1/2* 64* RS We know that the area of △ PQR is 1408. Let's substitute 1408 for A and solve the equation for RS.
We found that RS is 44 units long.
The perimeter of a polygon is the sum of all its side lengths. Let's write an equation for the perimeter of △ JKL.
P = JK + KL + LJ
From the diagram, we know that JK=30 and KL=44. However, we do not know the length of LJ. The good news is that we can find it by using the fact that the area of the triangle is 650 square units and that the height corresponding to LJ measures 26 units.
Let's substitute 650 for A, 26 for h, and LJ for b into the formula for the area of a triangle. Then we can solve for LJ.
Now we are ready to find the perimeter of △ JKL.
The perimeter of the triangle is 124 units.
security
report
code
Find the required information in each of the following diagrams.
Find the area of parallelogram ABCD.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"A=","formTextAfter":null,"answer":{"text":["6"]}}
Let PQRS be a parallelogram with PQ=3, QR=4.8, and whose area is 12.48.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"h_1+h_2=","formTextAfter":null,"answer":{"text":["6.8"]}}
Let JKLM be a trapezoid whose area is 14 and let NKLM be a parallelogram.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"P=","formTextAfter":null,"answer":{"text":["14"]}}
security
report
code
security
report
code
The area of a parallelogram is the product of its base and height. A = bh The base can be any side of the parallelogram and the height of the parallelogram is the perpendicular distance between the bases. We can see that the two heights of the parallelogram are already given in the diagram.
If we want to find the area correctly, it is important to use the correct data pair. The height is different for each base, so let's pair each base to its corresponding height.
| Base | Height |
|---|---|
| AB=4 | EC=1.5 |
| BC=2 | DF=3 |
We can find the area of parallelogram ABCD by using any of these two pairs of values. For simplicity, let's use the ones in the second row. A = bh ⇒ A &= 2* 3 ⇒ A &= 6 The area of the parallelogram is 6 square units. Using the values in the first row leads to the same result!
We are given the side lengths and area of a parallelogram. Let's identify the two possible heights. We will begin by drawing and labeling them.
We can determine the heights by using the formula for the area of a parallelogram again. A = bh If we want to find h_1, we will take PS as the base. Let's substitute PS=4.8 and A=12.48 into the formula.
Now let's find h_2. This time we will take PQ as the base, so let's substitute PQ=3 and A=12.48 into the formula.
We found the two heights of parallelogram PQRS. Let's add them and round to one decimal place. h_1+h_2 = 2.6+4.16 = 6.76 ⇓ h_1+h_2 ≈ 6.8 The sum of the two possible heights of parallelogram PQRS is about 6.8.
The perimeter of a polygon is the sum of its side lengths. Let's begin by writing an expression for the perimeter of parallelogram NKLM.
P = NK+KL+LM+MN
Since the opposite sides of a parallelogram are congruent, we can simplify the previous equation a bit.
P = 2KL+2LM
The first thing we notice is that we do not know any of the side lengths of the parallelogram. However, we know the area of trapezoid JKLM. Recall that the area of a trapezoid is half the product of the height and the sum of the length of the bases.
A_(JKLM) = 1/2h(b_1+b_2)
We also know the length of the longer base JK and the height of the trapezoid MJ. Let's substitute the given information into the formula and solve it for the shorter base LM.
Our next step is to find KL.
Notice that OP is an altitude of parallelogram NKLM corresponding to the base KL. If we knew the area of parallelogram NKLM, we could find the length of KL. A = KL* OP The good news is that we can find the area of the parallelogram! Notice that MJ is an altitude if we consider the base of the parallelogram to be LM. Let's use this information to find the area of NKLM.
Now that we know the area of the parallelogram, we can use it along with the height OP=1.6 to find KL. Let's do it!
We are finally ready to find the perimeter of the parallelogram.
The perimeter of the parallelogram is 14 units!
security
report
code
Find the required information for each figure.
Find the area of the following rhombus.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"A=","formTextAfter":null,"answer":{"text":["7.5"]}}
The area of the following rhombus is 24.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"PR=","formTextAfter":null,"answer":{"text":["8"]}}
Find the side length of the following rhombus. Round the answer to the nearest integer.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"JK\u2248","formTextAfter":null,"answer":{"text":["23"]}}
security
report
code
security
report
code
The area of a rhombus is half the product of the lengths of its diagonals. A = 1/2d_1 d_2 In our case, the diagonals are AC and BD.
From the diagram, we have that AC=5 and BD=3. Let's substitute these values to find the area of the rhombus.
The area of the rhombus is 7.5 units.
We are given the area and the length of one diagonal of a rhombus.
Notice that PR is the other diagonal of the rhombus. We can find its length by substituting 24 for A and 6 for d_1 into the formula for the area of a rhombus. Let's do it!
The second diagonal PR is 8 units long.
We need to find the side length of the given rhombus.
Since a rhombus is a parallelogram, we can use the formula for finding the area of a parallelogram to help us find the missing side length. The area of a parallelogram is the product of its base and height. From the diagram, we can see that NL is the height of the rhombus that corresponds to the base JK. A = bh ⇒ A = JK* NL We know NL but we do not know the area of the rhombus. However, remember that we can also find the area of a rhombus by using its diagonals. The area of a rhombus is half the product of the lengths of its diagonals. A = 1/2d_1 d_2 The diagonals measure 37 and 26 units. Let's substitute these values into the formula and find the area of the figure.
Now that we know the area of the rhombus, we can substitute it into the formula for the area of the parallelogram using the base and height to find JK. Let's do it!
The base of the rhombus, and therefore the side length of the figure, is about 23 units.
security
report
code
Find the requested information for each of the following diagrams.
Find the area of trapezoid ABCD.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"A=","formTextAfter":null,"answer":{"text":["204"]}}
The area of the following trapezoid is 220.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"RS=","formTextAfter":null,"answer":{"text":["29"]}}
The area of ABCD is 450 and the area of AECD is 292.5.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"AE=","formTextAfter":null,"answer":{"text":["9"]}}
security
report
code
security
report
code
The area of a trapezoid is half the height times the sum of the lengths of the bases. A = 1/2h(b_1+b_2) From the diagram, we can see that the trapezoid has bases 24 and 10 and that it has a height of 12.
Let's substitute the given information into the formula to find the area.
The area of the figure is 204 square units.
We are told that the trapezoid has an area of 220. Additionally, we know that the area of a trapezoid is half the height times the sum of the lengths of the bases.
A = 220 ⇒ 1/2h(b_1+b_2) = 220
The bases of the given trapezoid are the parallel sides PQ and RS. The height is the perpendicular segment between the bases, PT.
The height of the trapezoid is PT=10 and the length of the shorter base is PQ=15. Let's substitute this information into the equation we wrote before and solve for the longer base RS.
We found that the longer base RS is 29 units long.
We begin by noticing that AB∥CD and BC∥AD. This means that ABCD is a parallelogram, so its area is the product of its base and height.
If the base of the parallelogram is AB, the height is 15. We are told that the parallelogram has an area of 450. Let's substitute these values into the formula for the area to find AB.
The length of the base AB is 30 units. Since ABCD is a parallelogram, we can say that DC is also 30 units long. However, we want to find the length of AE. We can see in the diagram that AECD is a trapezoid and that AE is its smaller base. We can use the area of the trapezoid to find the missing base length. A=1/2h(b_1+b_2) We are given that the area of the figure is 292.5. The height of the trapezoid is the same as the height of the parallelogram, 15, and the two bases of the trapezoid are DC=30 and AE. Let's substitute these values into the equation and solve for AE.
The length of the shorter base AE is 9 units.
security
report
code
Areas of Parallelograms, Triangles, and Trapezoids
Recommended exercises
To get personalized recommended exercises, please login first.
Loading content
≤
>
■2
■▢
e▢
7
8
9
×
÷■1
=
=
√▢
▢√
∫▢▢
4
5
6
+
−
≤
<
log
ln
log▢
1
2
3
()
∣
sin
cos
tan
0
.
π
x
y
Loading content