Personal Blogs

This is something I found on Quora 

Above we see three regular polygons, the equilateral triangle, square and regular pentagon. Each has a side length of 2 units. Associated with any regular polygon are two circles: its incircle, which touches its sides, and its circumcircle, which passes through its vertices.

Surprisingly, the area between these two circles, shaded pink in the figures above, always has an area of π/4 square units, however many sides the regular polygon has. How do we know this?

Here we see two consecutive vertices of the polygon, painy=ts A and B, with M being the midpoint of AB and O the common centre of the incentre and the circumcentre.

The side length is 1, so MB is ½, and because AB is tangent to the incentre and OM a radius of the circle angle OMB is a right angle. Let the radiiusof the circumcentre be R and that of the incircle r, so OB = R and OM = r.

Then using Pythagoras in the irght-angled triangle OMB we have

OM² + (½)² = OM², so r² + ¼ = R², or R² - r² = ¼.

But now considering the ares of the two circles we have

Area of circumcircle - Area incircle = πR² - πr² = π/4.

Notice we haven't used the number of sides anywhere in this! So the area is always π/4 irrespective of the number of sides, as claimed.

I always wrongly suppose Pine Martens, animals related to weasels and stoats, are exclusively carnivorous  , like their cousins. But it seems now that they also eat fruit and other stuff and are particularly partial to raspberry jam sandwiches. 

A friend spotter this seal on the beach at Aldeburgh. I can’t embed the video on this page but here’s a link to a YouTube playloop:

https://youtube.com/shorts/DNM5P53iUfw?si=NUuBLHTgLb_ABzQk

All night long

The wind was roaring like a lion

About to gobble us up.

I’ve heard it said that there is no word in the English language that rhymes with ‘oblige’, which, as far as I can see is true, unless you count proper names, such as ‘Nige’.

So what about words that only have one other word that rhymes with them? Well, I thought of wombat and combat; but are there any other words that rhyme with these two? If you find any do write in the comments.

Being overcharged for a pencil eraser. That’s daylight rubbery.

I keep missing the first letter off words. The doctor says it's rain damage.

This is a tooth from a cave bear. My brother gave it to me some years ago and I keep it by my bedside, as a sort of link to the past and the world in which our ancestors lived. 

The coin is to give an indication of scale. It's a Roman denarius, about 1 cm across.

• The bears' diet was too specialised (think about giant pandas), the plants they eat were affected by climate change and there wasn't enough food
• Humans hunted them to extinction
• Humans out-competed them.