Circumference: Using additional geometric shapes

Below is a circle bounded by a parallelogram:

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

What is the area of the parallelogram?

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:

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

And 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.

Since the circumference is 25.13.

Circumference formula:$2\pi R$
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 you can calculate with a parallelogram area formula:

$AlturaXLado$

$9\times8\approx72$

$\approx72$

The area of the rectangle in the drawing is 28X cm².

What is the area of the circle?

Let's draw the center of the circle and we can divide the diameter of the circle into two equal radii

Now let's calculate the length of the radii as follows:

$7\times2r=28x$

$14r=28x$

We'll divide both sides by 14:

$r=\frac{28}{14}x$

$r=2x$

Let's calculate the circumference of the circle:

$P=2\pi\times r=2\pi\times2x=4\pi x$

$4\pi x$

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?

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:

$72-56.24\approx21.73$

$\approx21.73$

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

112

ABCD is a deltoid with an area of 58 cm².

DB = 4

AE = 3

What is the ratio between the circles that have diameters formed by AE and and EC?

3:26

Ivan does laps around a circular park which has a radius of 300 meters.

He completes 5 full circuits in 35 minutes.

What was Ivan's average speed?

269.14 meter per minute

The circumference of the circle in the diagram is $36a^2$ cm.

BO is the radius.

ABCD is a parallelogram.
BO is perpendicular to DC.

DC = $\frac{4}{a}$

What is the area of the parallelogram?

$72\frac{a}{\pi}$ cm²

Look at the triangle and circle below.

Which has the larger perimeter/circumference?

The circle

In the drawing, a rectangle and a circle whose center is the corner of the rectangle.

Given R=4

What is the length of the highlighted part in the drawing?

$\frac{32}{45}\pi$