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Mathematical Fundamentals

The essential tools for algebra, geometry, probability, and logic!

Engaging with Geometry


What is the area of the region shaded green?


Hint: There's a shortcut so that you don't have to count the green squares.

Is there more area shaded pink or blue?

As you saw in the previous two problems, strategy can be more important than formulas when solving problems. By cutting and/or rearranging pieces, we can compare areas intuitively and efficiently.

As you move through the next several problems, continue to use creative strategies and your intuition about shapes to find solutions.

Given that figures A, B, and C have the same dimensions, in which figure is the area of the regions shaded purple the greatest?

Note: The green quarter-circles of A, green half-circles of B, and green full circle of C all have the same radius.

What is the area of the red portion of the face?


Three sets of eight identical cubes are placed into the three different figures shown. Which figure has the greatest surface area?

Surface area is the total area of all of a 3D figure's surfaces.

27 unit cubes are packed together to create this \(3\times 3\times 3\) cube. Each face of a unit cube has an area of 1.

If all 8 of the corner cubes are removed, by how much will the surface area of the figure increase?


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