Calculate Parallelogram Area: Circle with Circumference 25.13 and Tangent Points

Q: Why are the tangent segments from each vertex equal?

This is a fundamental tangent property! When two tangent lines are drawn from the same external point to a circle, the segments are always equal. So from point A: AE = AF = 3, and from point B: BG = BF = 6.

Q: How do I find the parallelogram's height from the circle?

The height equals the diameter of the inscribed circle! Since all sides are tangent to the circle, the distance between parallel sides equals the diameter = 2r.

Q: How do I get radius 4 from circumference 25.13?

Use the circumference formula: $ C = 2\pi r $$ 25.13 = 2\pi r $$ r = \frac{25.13}{2\pi} \approx \frac{25.13}{6.28} \approx 4 $

Q: What if the parallelogram isn't drawn to scale?

Always rely on the given measurements, not the visual appearance! Use the tangent lengths (3 and 6) and circumference (25.13) to calculate, regardless of how the diagram looks.

Tangent Parallelogram Area with Circle Properties

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?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:11 Let's find the area of the parallelogram.

00:15 Lines starting from the same point are equal until they meet.

00:24 A side is equal to the sum of its parts.

00:34 Draw a diameter in the circle.

00:39 The radius is always at a right angle to the tangent where they meet.

00:44 We'll use the circle's circumference formula to find the radius.

00:49 Let's substitute the values and find the radius.

00:58 Round this radius. Now we know the radius and diameter!

01:05 To get the area, multiply the height, E H, by the side, A B.

01:11 Substitute the values and solve for the area.

01:14 And that's how we solve this problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

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?

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:

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

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:

$AlturaXLado$

$9\times8\approx72$

Final Answer

$\approx72$

Key Points to Remember

Essential concepts to master this topic

Tangent Property: Equal tangent segments from external point to circle
Technique: Use $C = 2\pi r$ to find radius from circumference 25.13
Check: Diameter equals height, area = base × height = 9 × 8 = 72 ✓

Common Mistakes

Avoid these frequent errors

Using circumference as diameter
Don't use circumference 25.13 directly as the diameter = area of 226! The circumference must be divided by π to get diameter. Always use C = 2πr to find radius first, then diameter = 2r.

Practice Quiz

Test your knowledge with interactive questions

Calculate the area of the parallelogram according to the data in the diagram.

FAQ

Everything you need to know about this question

Why are the tangent segments from each vertex equal?

This is a fundamental tangent property! When two tangent lines are drawn from the same external point to a circle, the segments are always equal. So from point A: AE = AF = 3, and from point B: BG = BF = 6.

How do I find the parallelogram's height from the circle?

The height equals the diameter of the inscribed circle! Since all sides are tangent to the circle, the distance between parallel sides equals the diameter = 2r.

Why is the base length 3 + 6 = 9?

The base AB consists of two tangent segments: AF = 3 and FB = 6. Since these segments are adjacent on the same side, we add them: AB = AF + FB = 3 + 6 = 9.

How do I get radius 4 from circumference 25.13?

Use the circumference formula: $C = 2\pi r$
$25.13 = 2\pi r$
$r = \frac{25.13}{2\pi} \approx \frac{25.13}{6.28} \approx 4$

What if the parallelogram isn't drawn to scale?

Always rely on the given measurements, not the visual appearance! Use the tangent lengths (3 and 6) and circumference (25.13) to calculate, regardless of how the diagram looks.

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Calculate Parallelogram Area: Circle with Circumference 25.13 and Tangent Points

Tangent Parallelogram Area with Circle Properties

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why are the tangent segments from each vertex equal?

How do I find the parallelogram's height from the circle?

Why is the base length 3 + 6 = 9?

How do I get radius 4 from circumference 25.13?

What if the parallelogram isn't drawn to scale?

🌟 Unlock Your Math Potential

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