Inscribed and circumscribed shapes are important tools used in real life. They help people in fields like architecture, engineering, and computer graphics to solve problems. By understanding how these shapes work with circles and polygons, we can create better designs and do calculations more efficiently.

1. Inscribed Figures:

• An inscribed polygon is a shape where all the corners touch the edge of a circle. For example, when city planners design roads, they may use inscribed polygons to make sure roundabouts fit nicely into the street plans.

• You can figure out the area of an inscribed polygon with this formula:

  A = (1/2) * n * r * sin(2π/n)

  Here, n is the number of sides, and r is the radius of the circle.

2. Circumscribed Figures:

• A circumscribed polygon is a shape that goes around a circle. All its sides touch the circle. Engineers often use this idea when making gears and wheels, making sure the circle fits perfectly around other structures.

• The area of a circumscribed shape is important when figuring out how to use materials wisely. The area can be calculated with:

  A = P * r

  In this formula, P is the perimeter (the distance around the shape), and r is the radius of the circle inside it.

3. Applications:

• Studies show that using circumscribed and inscribed shapes can help cut material costs by up to 20% in construction. They also improve design efficiency by allowing better spacing of parts.

By using what we know about inscribed and circumscribed shapes, we can make big improvements in design, make better use of resources, and work more efficiently.