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Rectangle calculator
Input two rectangle dimensions (length, width, diagonal, area and perimeter), and the calculator will calculate the unknown property. The calculator accepts all types of input values, including fractions and square roots, and provides step-by-step explanation.
Find the diagonal $ d $ of a rectangle if side $a = \frac{ 3 }{ 2 }$ and perimeter $P = \frac{ 11 }{ 2 }$.
solution
$$ d = \frac{\sqrt{ 61 }}{ 4 } $$explanation
STEP 1: find side $ b $
To find side $ b $ use formula:
$$ P = 2 a + 2 b $$After substituting $ P = \frac{ 11 }{ 2 } $ and $ a = \frac{ 3 }{ 2 } $ we have:
$$ \frac{ 11 }{ 2 } = 2 \cdot \frac{ 3 }{ 2 } + 2 b $$ $$ \frac{ 11 }{ 2 } = 3 + 2 b $$ $$ 2 b = \frac{ 11 }{ 2 } - 3 $$ $$ 2 b = \frac{ 5 }{ 2 } $$ $$ b = \dfrac{ \frac{ 5 }{ 2 } }{ 2 } $$ $$ b = \frac{ 5 }{ 4 } $$STEP 2: find diagonal $ d $
To find diagonal $ d $ use Pythagorean Theorem:
$$ a^2 + b^2 = d^2 $$After substituting $ a = \frac{ 3 }{ 2 } $ and $ b = \frac{ 5 }{ 4 } $ we have:
$$ \left(\frac{ 3 }{ 2 }\right)^2 + \left(\frac{ 5 }{ 4 }\right)^2 = d^2 $$ $$ \frac{ 9 }{ 4 } + \frac{ 25 }{ 16 } = d^2 $$ $$ d^2 = \frac{ 61 }{ 16 } $$ $$ d = \sqrt{ \frac{ 61 }{ 16 } } $$$$ d = \frac{\sqrt{ 61 }}{ 4 } $$
| $$ A = ab $$ |
| area |
| $$ P = 2a + 2b $$ |
| perimeter |
| $$ d^2 = a^2 + b^2 $$ |
| diagonal |
Rectangle calculations
This calculator uses the following formulas to find the missing values of a rectangle:
| Area: | $$ A = a \cdot b $$ |  |
| Perimeter: | $$ P = 2a + 2b $$ | |
| Diagonal: | $$ d^2 = a^2 + b^2 $$ |
Example 01 :
What is the area of a rectangle with a base of 12 cm and a height of 3/2 cm?
Solution:
base $ a = 6 $
height $ b = \dfrac{9}{2} $
$$ \color{blue}{A = a \cdot b} = 6 \cdot \frac{9}{2} = \frac{54}{2} = 27 $$Example 02 :
What is the perimeter of a rectangle with a length of 7/2cm and a width of 5/2cm?
Solution:
length $ a = \dfrac{7}{2} \, cm $
width $ b = \dfrac{5}{2} \, cm $
$$ \color{blue}{P = 2a + 2b} = 2 \cdot \frac{7}{2} + 2 \cdot \dfrac{5}{2} = 7 + 5 = 12 $$Example 03 :
The area of a rectangle is 42 cm2. Find its perimeter if the width is 7cm.
Solution:
We'll need two steps to solve this one:
Step 1: find length ( b ):
width $ a = 7 cm $
area: $ A = 42 cm $
$$ \begin{aligned} A & = a \cdot b \\[ 1 em] 42 & = 7 \cdot b \\[ 1 em] b & = \frac{42}{7}\\[ 1 em] b & = 6 \\[ 1 em] \end{aligned} $$Step 2: find perimeter ( P )
width $ a = 7 cm $
length $ b = 6 cm $
$$ \begin{aligned} P & = 2a + 2b \\[ 1 em] P & = 2 \cdot 7 + 2 \cdot 6 \\[ 1 em] P & = 14 + 12 \\[ 1 em] P & = 28 \, cm^2 \\[ 1 em] \end{aligned} $$Example 04 :
What is the diagonal of a rectangle if the perimeter is P = 11/2 cm and a width is a = 3/2 cm ?
Solution:
Step 1: find length ( b ):
width $ a = \dfrac{3}{2} cm $
perimeter: $ P = \dfrac{11}{2} cm $
$$ \begin{aligned} P & = 2a + 2b \\[ 1 em] \frac{11}{2} & = 2 \cdot \frac{3}{2} + 2b \\[ 1 em] \frac{11}{2} & = 3 + 2b \\[ 1 em] 2b &= \frac{11}{2} - 3 \\[1 em] 2b &= \frac{5}{2} \\[1 em] b &= \frac{5}{4} \end{aligned} $$Step 2: find diagonal ( d )
width $ a = \dfrac{3}{2} cm $
length $ b = \dfrac{5}{4} cm $
$$ \begin{aligned} d^2 & = a^2 + b^2 \\[ 1 em] d^2 & = \left( \frac{3}{2} \right)^2 + \left( \frac{5}{4} \right)^2 \\[ 1 em] d^2 & = \frac{9}{4} + \frac{25}{16} \\[ 1 em] d^2 & = \frac{61}{16} \\[ 1 em] d & = \frac{\sqrt{61}}{4} \end{aligned} $$Please tell me how can I make this better.