Area of a Trapezoid: Using additional geometric shapes

Question 1

The trapezoid DECB forms part of triangle ABC.

AB = 6 cm
AC = 10 cm

Calculate the area of the trapezoid DECB, given that DE divides both AB and AC in half.

Question 2

Given that the triangle ABC is isosceles,
and inside it we draw EF parallel to CB:

AF=5 AB=17
AG=3 AD=8
AD the height in the triangle

What is the area of the trapezoid EFBC?

Question 3

Given: the area of the triangle is equal to 2 cm² and the height of the triangle is 4 times greater than its base.

The area of the trapezoid is equal to 12 cm² (use x)

Calculate the value of x.

Question 4

Look at the isosceles trapezoid ABCD below.

DF = 2 cm
AD = $\sqrt{20}$ cm

Calculate the area of the trapezoid given that the quadrilateral ABEF is a square.

Question 5

In the drawing, a trapezoid is given, with a semicircle at its upper base.

The length of the highlighted segment in cm is $7\pi$

Calculate the area of the trapezoid

Examples with solutions for Area of a Trapezoid: Using additional geometric shapes

Exercise #1

The trapezoid DECB forms part of triangle ABC.

AB = 6 cm
AC = 10 cm

Calculate the area of the trapezoid DECB, given that DE divides both AB and AC in half.

Video Solution

Step-by-Step Solution

DE crosses AB and AC, that is to say:

$AD=DB=\frac{1}{2}AB=\frac{1}{2}\times6=3$

$AE=EC=\frac{1}{2}AC=\frac{1}{2}\times10=5$

Now let's look at triangle ADE, two sides of which we have already calculated.

Now we can find the third side DE using the Pythagorean theorem:

$AD^2+DE^2=AE^2$

We substitute our values into the formula:

$3^2+DE^2=5^2$

$9+DE^2=25$

$DE^2=25-9$

$DE^2=16$

We extract the root:

$DE=\sqrt{16}=4$

Now let's look at triangle ABC, two sides of which we have already calculated.

Now we can find the third side (BC) using the Pythagorean theorem:

$AB^2+BC^2=AC^2$

We substitute our values into the formula:

$6^2+BC^2=10^2$

$36+BC^2=100$

$BC^2=100-36$

$BC^2=64$

We extract the root:

$BC=\sqrt{64}=8$

Now we have all the data needed to calculate the area of the trapezoid DECB using the formula:

(base + base) multiplied by the height divided by 2:

Keep in mind that the height in the trapezoid is DB.

$S=\frac{(4+8)}{2}\times3$

$S=\frac{12\times3}{2}=\frac{36}{2}=18$

Answer

Exercise #2

Given that the triangle ABC is isosceles,
and inside it we draw EF parallel to CB:

AF=5 AB=17
AG=3 AD=8
AD the height in the triangle

What is the area of the trapezoid EFBC?

Video Solution

Step-by-Step Solution

To find the area of the trapezoid, you must remember its formula: $\frac{(base+base)}{2}+\text{altura}$ We will focus on finding the bases.

To find GF we use the Pythagorean theorem: $A^2+B^2=C^2$ In triangle AFG

We replace:

$3^2+GF^2=5^2$

We isolate GF and solve:

$9+GF^2=25$

$GF^2=25-9=16$

$GF=4$

We will do the same process with side DB in triangle ABD:

$8^2+DB^2=17^2$

$64+DB^2=289$

$DB^2=289-64=225$

$DB=15$

From here there are two ways to finish the exercise:

Calculate the area of the trapezoid GFBD, prove that it is equal to the trapezoid EGDC and add them up.
Use the data we have revealed so far to find the parts of the trapezoid EFBC and solve.

Let's start by finding the height of GD:

$GD=AD-AG=8-3=5$

Now we reveal that EF and CB:

$GF=GE=4$

$DB=DC=15$

This is because in an isosceles triangle, the height divides the base into two equal parts then:

$EF=GF\times2=4\times2=8$

$CB=DB\times2=15\times2=30$

We replace the data in the trapezoid formula:

$\frac{8+30}{2}\times5=\frac{38}{2}\times5=19\times5=95$

Answer

Exercise #3

Given: the area of the triangle is equal to 2 cm² and the height of the triangle is 4 times greater than its base.

The area of the trapezoid is equal to 12 cm² (use x)

Calculate the value of x.

Video Solution

Answer

$x=2$

Exercise #4

Look at the isosceles trapezoid ABCD below.

DF = 2 cm
AD = $\sqrt{20}$ cm

Calculate the area of the trapezoid given that the quadrilateral ABEF is a square.

Video Solution

Answer

Exercise #5

In the drawing, a trapezoid is given, with a semicircle at its upper base.

The length of the highlighted segment in cm is $7\pi$

Calculate the area of the trapezoid

Video Solution

Answer

112

Question 1

ABCD is a right-angled trapezoid

Given AD perpendicular to CA

BC=X AB=2X

The area of the trapezoid is $\text{2}.5x^2$

The area of the circle whose diameter AD is $16\pi$ cm².

Find X

Question 2

ABCD is a kite

ABED is a trapezoid with an area of 22 cm².

AC is 6 cm long.

Calculate the area of the kite.

Question 3

The tapezoid ABCD and the parallelogram ABED are shown below.

EBC is an equilateral triangle.

What is the area of the trapezoid?

Question 4

ABCD is a trapezoid.

$\frac{2}{7}=\frac{EA}{ED}$

What is the area of the trapezoid?

Question 5

The trapezoid ABCD is placed on top of the square CDEF square.

CDEF has an area of 49 cm² .

What is the trapezoidal area?

Exercise #6

ABCD is a right-angled trapezoid

Given AD perpendicular to CA

BC=X AB=2X

The area of the trapezoid is $\text{2}.5x^2$

The area of the circle whose diameter AD is $16\pi$ cm².

Find X

Video Solution

Answer

4 cm

Exercise #7

ABCD is a kite

ABED is a trapezoid with an area of 22 cm².

AC is 6 cm long.

Calculate the area of the kite.

Video Solution

Answer

$6\sqrt{13}$ cm²

Exercise #8

The tapezoid ABCD and the parallelogram ABED are shown below.

EBC is an equilateral triangle.

What is the area of the trapezoid?

Video Solution

Answer

$27.3$ cm².

Exercise #9

ABCD is a trapezoid.

$\frac{2}{7}=\frac{EA}{ED}$

What is the area of the trapezoid?

Video Solution

Answer

$45$ cm².

Exercise #10

The trapezoid ABCD is placed on top of the square CDEF square.

CDEF has an area of 49 cm² .

What is the trapezoidal area?

Video Solution

Answer

$18$ cm²

Question 1

Trapezoid ABCD is enclosed within a circle whose center is O.

The area of the circle is $16\pi$ cm².

What is the area of the trapezoid?

Question 2

The right-angled trapezoid ABCD is shown below.

ABED is a parallelogram.

Calculate the area of the trapezoid.

Question 3

From the point O on the circle we take the radius to the point D on the circle. Given the lengths of the sides in cm:

DC=8 AE=3 OK=3 EK=6

EK is perpendicular to DC

Calculate the area between the circle and the trapezoid (the empty area).

Question 4

ABC is a right triangle.

DE is parallel to BC and is the midsection of triangle ABC.

BC = 5 cm

AC = 13 cm

Calculate the area of the trapezoid DECB.

Exercise #11

Trapezoid ABCD is enclosed within a circle whose center is O.

The area of the circle is $16\pi$ cm².

What is the area of the trapezoid?

Video Solution

Answer

$22.75$ cm².

Exercise #12

The right-angled trapezoid ABCD is shown below.

ABED is a parallelogram.

Calculate the area of the trapezoid.

Video Solution

Answer

$40$ cm²

Exercise #13

From the point O on the circle we take the radius to the point D on the circle. Given the lengths of the sides in cm:

DC=8 AE=3 OK=3 EK=6

EK is perpendicular to DC

Calculate the area between the circle and the trapezoid (the empty area).

Video Solution

Answer

36.54

Exercise #14

ABC is a right triangle.

DE is parallel to BC and is the midsection of triangle ABC.

BC = 5 cm

AC = 13 cm

Calculate the area of the trapezoid DECB.

Video Solution

Answer

22.5