If the revolved curve is not a circle, the surface is called a toroid, as in a square toroid. If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere. If the axis of revolution passes twice through the circle, the surface is a spindle torus (or self-crossing torus or self-intersecting torus). If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution, also known as a ring torus. A ring torus is sometimes colloquially referred to as a donut or doughnut. The main types of toruses include ring toruses, horn toruses, and spindle toruses. In geometry, a torus (plural tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle. Now that we've done that, we can solve a similar problem, where instead of a square inscribed in a circle, we have a circle inscribed in a square.A ring torus with aspect ratio 3, the ratio between the diameters of the larger (magenta) circle and the smaller (red) circle. a 2/8.Īnd if we have the radius, A shaded=(A circle-A square)/4= (π.If we have the side of the square, a, we get A shaded=(A circle-A square)/4=(π So the shaded area is A shaded=(A circle-A square)/4 The sum of their areas is the difference between the area of the circle and the area of the square. Here's it is very easy - the 4 irregular shapes are all the same size (from symmetry). The strategy for finding the area of irregular shapes is usually to see if we can express that area as the difference between the areas formed by two or more regular shapes. Now that we've done this, we can apply our knowledge to solve various kinds of "find the area of the shaded shape" problems related to a square inscribed in a circle, like this one: Problem 3Ī square with side a is inscribed in a circle. We've already seen how to find the length of a square's diagonal from its side: it is a We already have the key insight from above - the diameter is the square's diagonal. Find formulas for the circle's radius, diameter, circumference and area, in terms of a. Problem 2Ī square with side a is inscribed in a circle. Now let's do the converse, finding the circle's properties from the length of the side of an inscribed square. √2 ( Pythagorean theorem applied to a 45-45-90 triangle), the area is then 2r 2, and the perimeter is 4.So the central angle measures 180°, which means it is the diameter.Īrmed with this knowledge, the length of the square's diagonal is simply 2r, each side measures r Since these angles are inscribed angles in a circle, they measure half of the central angle on the same arc. We can show this using a symmetry argument - the square is symmetrical across its diagonal, so the diagonal must pass through the center of the circle.Īlternatively, we know that the square's interior angles are all right angles, which measure 90°. The key insight to solve this problem is that the diagonal of the square is the diameter of the circle. Find formulas for the square's side length, diagonal length, perimeter and area, in terms of r. Problem 1Ī square is inscribed in a circle with radius 'r'. When a square is inscribed in a circle, we can derive formulas for all its properties- length of sides, perimeter, area and length of diagonals, using just the circle's radius.Ĭonversely, we can find the circle's radius, diameter, circumference and area using just the square's side.
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