This page covers ratio of areas inscribed shapes, a challenging and advanced topic from the book "Advanced ACT Math". The problems here focus on 2D and 3D figures where shapes are inscribed in circles, spheres, and other geometric forms. Understanding these concepts is crucial for students aiming for top scores, as they appear frequently on the test and are often not covered in detail in other materials.
The exercises include squares and rectangles inscribed in circles, right triangles with hypotenuses as diameters, and cubes inside spheres, along with methods for calculating areas, Ratio of Areas in Inscribed Shapes, and volumes. These problems are designed to push students beyond standard practice and develop strong problem-solving skills.
All of this material is taken directly from my ACT Math Problems Book, providing a reliable and structured source for advanced preparation. You can learn more about the book on my site ACT math book or on Amazon here.
If a rectangle is inscribed in a circle, the diameter of the circle equals the diagonal of the rectangle (note that a square is a special case of a rectangle). Similarly, if a right triangle is inscribed in a circle, the diameter of the circle equals the hypotenuse of the triangle.
For example, consider a square inscribed in a circle with an area of $100\pi$ square units. To find the area of the square, first determine the radius of the circle: $$ \pi r^2 = 100\pi \implies r^2 = 100 \implies r = 10. $$ Since the diagonal of the square corresponds to the diameter of the circle, the diagonal length is $2r = 20$. Using the properties of a $45^\circ$-$45^\circ$-$90^\circ$ triangle, the Pythagorean Theorem, or trigonometry, the side length $s$ is $\dfrac{20}{\sqrt{2}}= 10\sqrt{2}$. Thus, the area of the square is $(10\sqrt{2})^2=200$.
A frequent variation involves a circle inscribed in a square. This concept may also be framed in terms of multiple circles or cylindrical cans arranged within a box.
For instance, consider a circle with radius $r=5$ inscribed in a square. To find the area of the region inside the square but outside the circle, first note that the diameter of the circle is $2(5)=10$. This diameter equals the side length of the square. The area of the square is $10^2=100$, and the area of the circle is $\pi(5^2)=25\pi$. Therefore, the area of the region between the circle and the square is $100-25\pi$.
Show correct answer
Show correct answer
Show correct answer
Show correct answer
Show correct answer
Show correct answer
Show correct answer
Show correct answer
Suppose a cube is inscribed in a sphere. The long diagonal of the cube is equal to the diameter of the sphere ($d=2r$). Since the long diagonal of a cube with side length $s$ is $s\sqrt{3}$ (a relationship derived using the Pythagorean Theorem twice), the side length is equal to the diameter divided by $\sqrt{3}$. Thus, $s = \dfrac{2r}{\sqrt{3}}$.
Show correct answer
Show correct answer
Answer key: 1.B, 2.C, 3.A, 4.A, 5.A, 6.D, 7.C, 8.D, 9.C, 10.B
Mastering ratio of areas inscribed shapes is essential for achieving high scores in ACT Math, especially when dealing with advanced geometry problems. By understanding the relationships between shapes and their circumscribed figures, students can solve complex questions more efficiently. Consistent practice with these problems will significantly improve both accuracy and speed on the test.
