Areas

Area of a circle

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π is defined to be the ratio of a circle's circumference to its diameter. Let C denote the circumference, d the diameter, and r the radius: then \[\pi = \frac{C}{d}=\frac{C}{2r}\]

This ratio is the same for all circles regardless of the size.

Since \(C=2r\cdot \pi\), half of the circumference of a circle is given by \[\frac{C}{2}=r\cdot \pi\] This is the length of one of the sides of the rectangle in the worksheet above, the other side has the length r. By cutting up the circle in small circle sectors, and rearranging the sectors, one can se that the sectors almost fit in the rectangle. The thinner the sectors are, the better the fit.

Polygon Areas

Area of a triangle

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It is easy to see that the area of a right-angled triangle can be found through the area of a rectangle. This is shown by the green triangle/rectangle in the worksheet above. The sides of the rectangle are the same as the right-angled sides of the triangle. To find the area of the triangle, find the area of the rectangle and divide by two.

If all angles of a triangle are acute (less than 90°), you can make a rectangle as shown by the red triangle/rectangle above. In this case, one of the sides of the rectangle is the same as one of the sides of the triangle. The other side of the rectangle is the perpendicular height of the triangle. To find the area of the triangle, again, find the area of the rectangle and divide by two. Using the letters as in the worksheet above, the area A of the red triangle is \(A=\dfrac{b\cdot h}{2}\).

If one the angles of the triangle is obtuse (larger than 90°), the area of the triangle is half of the area of a parallelogram. In order to find the area of the triangle, find the area of the parallelogram and divide by two.

Area of a parallelogram

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You can choose any side as the base, then draw a perpendicular height, as shown below.

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A parallelogram can always be rearranged as a rectangle by cutting it to pieces. The rectangle has one side that is equal to one of the sides of the parallelogram, the base b. You get the height h of the rectangle by drawing a height perpendicular to the base of the parallelogram.

The area A of the large rectangle in the worksheet above, is given by \(A=(b+x)\cdot h=b\cdot h+x\cdot h\). The area of the parallelogram is given by the expression \(b\cdot h\).

Area of a trapezium

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If you have a trapezium, you can always place it inside a parallelogram, as shown in the worksheet above.

If the parallel sides of the trapezium are called a and b, and if the perpendicular height is called h, the parallelogram will have one side of length \(a+b\) and the perpendicular height h. The area of the parallelogram will hence be \((a+b)\cdot h\)

The area A of the trapezium is half of the area of the parallelogram, the area of the trapezium is given by \[A=\frac{(a+b)\cdot h}{2}=\left( \frac{a+b}{2}\right)\cdot h \]

Summary

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When it comes to the parallelogram and the triangle, it doesn't matter which side is chosen as the base. Check the Swap checkbox to see another side picked as the base. Note that the height may be drawn to a point outside of the polygon.

If you want to find the area of general polygons, you can find it by dividing the polygon into triangles.