Problems
Below is a regular octagon. Given that its side length is \(1\), what’s the difference between the area of the red rectangle and the rest of the octagon?
In the diagram, all the small squares are of the same size. What fraction of the large square is shaded?
Lines \(AB\) and \(CDE\) are parallel. Which triangle out of \(\triangle ABC\), \(\triangle ABD\) and \(\triangle ABE\) has the greatest area?
We ‘typically’ use the formula \(\frac{1}{2}bh\) for the area of a triangle, where \(b\) is the length of the base, and \(h\) is the perpendicular height. Here’s another one, called Heron’s formula.
Call the sides of the triangle \(a\), \(b\) and \(c\). The perimeter is \(a+b+c\). We call half of this the semiperimeter, \(s=\frac{a+b+c}{2}\). Then the area of this triangle is \[\sqrt{s(s-a)(s-b)(s-c)}.\] Prove this formula is correct.
We have a square of side length 1. At each vertex of the square, we draw a circle of radius 1. What is the area bounded by all four circles?
A simple polygon is a polygon that does not intersect itself and has no holes. Suppose we have a simple polygon \(S\) whose vertices consists of only integer coordinates.
The area turns out to be remarkably easy to calculate. Count up the number of points with integer coordinate inside the polygon and on the boundary; call them \(i\) and \(b\) respectively. The area is then \[A(S) = i+\frac{b}{2}-1.\]
In the picture above, \(i=3\) and \(b=11\), so \(A(S) = \frac{15}{2}\). Prove that this formula for the area \(A(S)\) is correct.
All of the rectangles in the figure below, which is drawn to scale, are similar to the big rectangle (that is, their sides are in the same ratio). Each number represents the area of the rectangle. What is the length \(AB\)?
