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Imagine a point $P$ moving along the unit circle from $0$ to $\pi/2$. It has a speed in the $x$-direction, and a speed in the $y$-direction (I'll call these $x$-speed and $y$-speed). Note that the $x$-speed of $P$ is constantly changing, and this is reflected in the graph that it draws (the unit circle): we are drawing the $y$-position, but with a changing $x$-speed.

Whereas, the sine graph is drawing the $y$-position with respect to the angle: this is a constant speed. Near an angle of $0$, the point $P$ has a very small $x$-speed, but the sine graph already has a significant $x$-speed. This results in the sine graph covering more area near the start of the movement of $P$ than $P$ does. Thus, the area under the sine wave is bigger than the area under the unit circle.

You could imagine taking the top half of the unit circle and stretching it at the ends where it hits the $x$-axis, to compensate for the slow $x$-speed, and turning it into the sine graph.