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math9maniac

Member

From: Tema

Registered: 2015-03-30

Posts: 443

Geometry question - rectangle

Hello, how y'all doing? I'd like some assistance on this.

What is the formula to use to find the area of a rectangle using the diagonals, not the length and breadth? I realised [half × product of diagonals] doesn't work since the diagonals don't intersect at right angles.
Are the diagonals of a rectangle always the same length?

Here's the actual question: Find the areaof a rectangle with both diagonals y mm long.

Many thanks in advance.

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Relentless

Member

Registered: 2015-12-15

Posts: 631

Re: Geometry question - rectangle

Hi, you can use Pythagoras' theorem to find the length of the diagonal in terms of the length and breadth. Answering your question requires manipulating the two formulas A = l*b and D = sqrt(l^2 + b^2)

Edit: I may have jumped to conclusions. I am having a bit of trouble using the algebra to get anything in terms of just D

Last edited by Relentless (2016-01-25 18:16:25)

Nehushtan

Member

Registered: 2013-03-09

Posts: 957

Re: Geometry question - rectangle

What is the formula to use to find the area of a rectangle using the diagonals, not the length and breadth?

You can't determine the area from just the diagonals alone. Two different rectangles may have different areas but the same diagonal length. For example, a rectangle with sides 3 and 4 and one with sides 2 and √21 both have diagonals of length 5 but their areas are clearly different.

Are the diagonals of a rectangle always the same length?

Yes.

You can however determine the area of a square from its diagonal. Maybe this is what you are thinking of?

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Relentless

Member

Registered: 2015-12-15

Posts: 631

Re: Geometry question - rectangle

Hi Nehushtan,

Thank you for putting a stop to my foolish play with algebra

Bob

Administrator

Registered: 2010-06-20

Posts: 10,623

Re: Geometry question - rectangle

hi math9maniac

Rectangles have two degrees of freedom, meaning you can choose the length, and then, independently, choose the width.  So you'll never find a formula with only one variable.

These rectangles have the same diagonals, but very different areas.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Relentless

Member

Registered: 2015-12-15

Posts: 631

Re: Geometry question - rectangle

Can we narrow down the range of areas for a given diagonal?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,623

Re: Geometry question - rectangle

I'll call one diagonal AC.

By choosing a point, B,  on the circle, you can make a new rectangle.  Choose it very close to A and the area is close to zero, so a lower bound is 0.

The upper bound is when the rectangle becomes a square.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Relentless

Member

Registered: 2015-12-15

Posts: 631

Re: Geometry question - rectangle

I see. So the final answer to the question is:

math9maniac

Member

From: Tema

Registered: 2015-03-30

Posts: 443

Re: Geometry question - rectangle

Thanks a lot you all for this assistance. So conclusion, it's utterly impossible using the diagonals alone.

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Only a friend tells you your face is dirty.

Nehushtan

Member

Registered: 2013-03-09

Posts: 957

Re: Geometry question - rectangle

You might want to try this additional problem:

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