Area of a Circle: Using additional geometric shapes

Question 1

The following is a circle enclosed in a parallelogram:

All meeting points are tangent to the circle.
The circumference is 25.13.

What is the area of the zones marked in blue?

Question 2

Below is an isosceles triangle drawn inside a circle:

What is the area of the circle?

Question 3

AD is perpendicular to BC

AD=3

The area of the triangle ABC is equal to 7 cm².

BC is the diameter of the circle on the drawing

What is the area of the circle?
Replace $\pi=3.14$

Question 4

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 5

Given the deltoid ABCD and the circle that its center O on the diagonal BC

Area of the deltoid 28 cm² AD=4

What is the area of the circle?

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

Exercise #1

The following is a circle enclosed in a parallelogram:

All meeting points are tangent to the circle.
The circumference is 25.13.

What is the area of the zones marked in blue?

Video Solution

Step-by-Step Solution

First, we add letters as reference points:

Let's observe points A and B.

We know that two tangent lines to a circle that start from the same point are parallel to each other.

Therefore:

$AE=AF=3$
$BG=BF=6$

From here we can calculate:

$AB=AF+FB=3+6=9$

Now we need the height of the parallelogram.

We know that F is tangent to the circle, so the diameter that comes out of point F will also be the height of the parallelogram.

It is also known that the diameter is equal to two radii.

It is known that the circumference of the circle is 25.13.

Formula of the circumference: $2\pi R$
We replace and solve:

$2\pi R=25.13$
$\pi R=12.565$
$R\approx4$

The height of the parallelogram is equal to two radii, that is, 8.

And from here it is possible to calculate the area of the parallelogram:

$\text{Lado }x\text{ Altura}$ $9\times8\approx72$

Now, we calculate the area of the circle according to the formula: $\pi R^2$

$\pi4^2=50.26$

Now, subtract the area of the circle from the surface of the trapezoid to get the answer:

$72-56.24\approx21.73$

Answer

$\approx21.73$

Exercise #2

Below is an isosceles triangle drawn inside a circle:

What is the area of the circle?

Video Solution

Answer

Exercise #3

AD is perpendicular to BC

AD=3

The area of the triangle ABC is equal to 7 cm².

BC is the diameter of the circle on the drawing

What is the area of the circle?
Replace $\pi=3.14$

Video Solution

Answer

17.1 cm².

Exercise #4

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 #5

Given the deltoid ABCD and the circle that its center O on the diagonal BC

Area of the deltoid 28 cm² AD=4

What is the area of the circle?

Video Solution

Answer

$49\pi$ cm².

Question 1

Given the triangle ABC when the base BC a semi-circle is drawn

The radius of the circle is equal to 3 cm and its center is the point D

Given AE=3 ED

What is the area of the dotted shape?

Question 2

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).

Exercise #6

Given the triangle ABC when the base BC a semi-circle is drawn

The radius of the circle is equal to 3 cm and its center is the point D

Given AE=3 ED

What is the area of the dotted shape?

Video Solution

Answer

$36-\text{4}.5\pi$ cm².

Exercise #7

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