# Radius of an arch and Pythagoras theorem

A door of width 6 meter has an arch above it having a height of 2 meter. Find the radius of the arch I analysed the problem to calculate the radius of curvature and I could not establish the relation between radius and height. Please someone help me to solve this using Pythagoras theorem on right angled triangle

Hint: Draw a picture The radius of the arch is $r=2+h$

• How can I calculate the value of h – Achari S Ganesha Jul 8 '14 at 18:36
• Is h=1.25cm and r=3.25cm – Achari S Ganesha Jul 8 '14 at 18:47
• That is close. The units of the problem are meters, not centimeters. – Ross Millikan Jul 8 '14 at 19:45
• If you extend the vertical line into a diameter, then you have $2 \times (2r-2) = 3 \times 3$ by power of a point. – Hao Ye Jul 9 '14 at 7:39

$r=2+h$
Also, \begin{align*} r^2 & =h^2+3^2\\ (2+h)^2 & = h^2+3^2 \end{align*} Solving we get $h = 1.25~\text{m}$, so radius $=2+h=2+1.25= 3.25~\text{m}$

• Here is a brief introduction to using $\LaTeX$ in your posts to create mathematical expressions. It has links to more extensive documentation, including the homegrown MathJax Basic Tutorial and Quick Reference. – hardmath Aug 24 '15 at 18:35

If radius is R, use property of circle of constant segments length product.

$$2 ( 2 R -2) = 3*3 \rightarrow R = \frac{13}{4}= 3.25$$