A circle with an inscribed square – can you find the area of the square?

Help with Middle and High School Math
Test Prep for High School Math, College Math, Teacher Certification Math and More!

Popular Math Courses:

Math Foundations

Math Skills Rebuilder Course:

Pre-Algebra

Algebra

Geometry

Algebra 2

Pre-Calculus

• MIDDLE & HIGH SCHOOL MATH
• HOMESCHOOL MATH
• COLLEGE MATH
• TEST PREP MATH
• TEACHER CERTIFICATION TEST MATH

[JoLowden-ozno]

Thank you for the videos. You are awesome at explaining everything.

[gavindeane]

A couple of ways...

The diagonal of the square = the diameter of the circle = 8

From a bit of Pythagoras (or from knowing that the sine and cosine of 45 degrees is 1/√2), the diagonal of a square is √2 × the side length, so the side length is 8/√2

So area = (8/√2)² = 64/2 = 32

Or...

Drawing the diagonal of the square makes two triangles, each with base length = diameter of the circle = 8 and triangle height = radius = 4.

Triangle area = ½ × base × height = ½×8×4 = 16. But we have two such triangles so total area = 16+16 = 32.

[panlomito]

This one is really easy again... rule of Pythagoras a² + b² = c² with for the square a=b
so the area of the square A = a² and diagonal of the square c = 2r = 8
a² + a² = 8² so 2a² = 64 and a² = 64/2 = 32 and that is the area of the square.

[tamjarvie]

Area of triangle = .5 x base x height = 4 x4 x2 = 32

[argonwheatbelly]

32 squ. in about 10 seconds in my head. It's a circumscribed square, so finding the area of one the right triangles made by drawing lines from the center to a corner of the square is easy enough. Actually, the area is 2r², where r is the radius of the circle.

[hexbinoban]

2 x radius = the diameter of the circle = 8
The diameter of the circle = one of the 2 diagonals of the square = 8
The diagonal split the square in 2 congruent right angle triangles with angles 45° 45° 90° --->
1 : 1 :√2
given :
Hypotenuse = 8
Calculate the other side --> 8 / √2 since it's a 45° 45° 90° with the sides ratio of 1 : 1 : √2
the length of the sides are 8 / √2 : 8 / √2 : 8
The area of a square is side x side = (8 / √2 ) x (8 / √2) = (8 / √2 )² = (64 / 2) = 32 some square units

[dazartingstall]

Come to think of it, if we draw two diagonals we have four right-angled triangles, each with base 4 and height 4, so:
4(2 × 4) = 32

[chrisdissanayake]

Let’s draw 2 lines, each joining the opposite corners of the square, passing through the center of the circle.

Then, the square has 4 right angled triangles inside it.
Each of the right angled triangle has two equal sides of 4 units each.
In a right angled triangle,
a^2+b^2= c^2,
where a and b are two sides and c is the hypotenuse.

a = b = 4

So, a^2 = 4^2= 16
b^2 = 4^2 = 16

a^2+b^2= 16+16=32
Therefore, c^2=32

c = each side of the square

Area of the square = c^2 = 32 units.

[johnleeson]

I didn't even think of a^2 + b^2 = c^2.

I got it this way:

Since the other two angles are 45 degrees, I took the SINE of 45 degrees and multiplied it by 8 and got 5.65685..., squared it, and got 32. The COSINE works, too.

Yep, overthinking a problem as usual...

[abinam]

Pssst .... too much talk and too much explanation causes confusion. You only needed about 2 minutes to adequately explain the method and solution.

[petersearls]

An easy way to calculate it is as follows.
1. Draw two diagonals creating 4 small triangles.
2. Label the top edge of the square C and the two legs of the small triangle, A and B.
3. We know that A=4. B=4
4. Using a^2 + b^2 = c^2
5. 4^2 + 4^2 = c^2 = 32
6. We also know that c^2 is the area of the square. Area = 32.

[thegorillaguide]

This sound much more complicated than it is. Using the KISS principle (Keep It Simple Stupid) Here's the simplest explanation of the simplest solution -

If a radius is drawn from the centre of the circle to each corner, the square becomes divided into four right-angle triangles each with two sides of 4. These can be simply rearranged into a rectangle with an area of 4 x 8. Job done.

[dazartingstall]

Diagonal of the square = 2r = 8
Let the side of the square be x and the area of the square be a.
2x² = 8²
2x² = 64
x² = 32
∴ a = 32 units²

[KaB-dn]

2r^2 should do it, unless I'm missing something.

[Ayelmar]

At the thumbnail, solved in my head, haven't gone back to check on paper or with calculator, I think the answer is 32 units.
My thinking:
The radius of the circle is 4, so the diagonal of the inscribed square is twice that, 8.
Working the Pythagorean theorem backwards for a triangle forming half the square, and plugging 8 in for C, we get c^2 = 64.
A^2 and B^2 are each half of C^2, or 32.
We could then get the square root of 32 for A & B, which would be but htis isn't necessary, since we can figure the area of the square as A * B, or in this case A^2 or B^2 since they're equivalten, so the result takes us back to 32.

[danharold]

The radius can be used as 2 known sides of a triangle with one side of the square being the 3rd.
4² + 4² = side² 32 = side²
we know that the area of a square is the side squared so we have the answer 32

[MrMousley]

If you draw a line from each corner of the square to the opposite corner
you then have four triangle shapes that can simply be arranged to make two squares
and these two squares are both 4 x 4 4 x 4 = 16 and 16 x 2 = 32
The square has an area of 32 sq cm

[tomtke]

the square's CORNERS touch the circle. Therefore the square's DIAGONAL = 2r.

A square's diagonal is 45° to its side. Therefore the 45/45/90 triangle has sides of ratio: 1/1/sqrt(2). Or the square's side
= (1/sqrt(2))×Diagonal
= (1/sqrt(2))×2R
= (1/sqrt(2))×8

Area of square
= S × S
= [(1/sqrt(2))×8]^2
= (1/2)(64)
= 32 units^2

[gopherspace]

OMG great lesson 👍
Wow 😂 thanks Mr J 👍👏🙏💪😎🌎

[DavidRobinson-rjsp]

Diameter = 8 = hypotenuse.

Square so bisected angles are 45.

8 Cos 45 = 4 root 2.

4 root 2 squared = 32.


Using COS is quicker.