Compute the *exact* area or length as directed. Square roots do not need to be simplified, but arithmetic does.

- Consider a heart shape (a rotated square with two semicircles). The side length of the square is 6, what is the area of the heart?

\( 36 + 9 \pi \) - Two pizzas at a restaurant have diameters of 6 and 12 inches, respectively. Little Timmy thinks this means the areas of the pizza are doubled? Do you agree or disagree with Timmy? Why?

Disagree. Area of smaller pizza is \( \pi 3^2 = 9\pi \). Area of larger pizza is \( \pi 6^2 = 36\pi \). The larger pizza is 4 times the area of the smaller pizza! - What is the height of an isosceles trapezoid with area of 10 units
^{2}, the smaller base has length 2, and the other base is 50% bigger?

\( 4 \) - What area of an equilateral triangle with side lengths measuring 8?

\( \text{triangle height} = \sqrt{48} = 4 \sqrt{3} \\ \text{triangle area} = \dfrac{1}{2} 8 \cdot \sqrt{48} = 4\sqrt{48} = 16 \sqrt{3} \) - What is the perimeter and area of an isosceles right triangle with hypotenuse of length 10?

\( \text{triangle legs} = \sqrt{50} = 5 \sqrt{2} \\ \text{perimeter} = 10 + 2\sqrt{50} = 10 + 10\sqrt{2}\) \\ \text{area} = 25 \) - A student computes the area an equilateral pentagon in the shape of a house (square with triangle on top). Each side length is 5. Do you agree with this student's answer? Why or why not?
\( \text{square's area: } 5^2 = 25 \)

\( \text{triangle's area: } \frac{1}{2} 5 \cdot 5 = \frac{25}{2} \) → Student used slant height \((5)\), not vertical height \( \left( \sqrt{5^2 - (5/2)^2} \right) \)

\( \text{total area: } 25 + \frac{25}{2} = \frac{75}{2} \)

correct area: \( 25 + \frac{5}{2} \sqrt{18.75} = 25 + \dfrac{5\sqrt{75}}{4} = 25 + \dfrac{25\sqrt{3}}{4} \)

- What is the area of a circle sector with radius 3 and degree measure of 120?

\( 3 \pi\) - What is the area of a circle sector with radius 4 and degree measure of 135?

\( 6 \pi\) - What is the area of a silo (rectangle with semicircle on top) where the height of the shape is 10 and the width is 4?

\(32 + 2\pi\) - What is the area of a flattened ice cream cone shape (isosceles triangle with semicircle on base) with total height of 10 and diameter of 4?

\( 2\pi + 16\) - Consider a shape with an equilateral triangle on top of a rectangle. The height of the entire shape is 10 units. The width of the shape is 4 units. What is the exact area of the shape?

\( 40 - 2\sqrt{12} = 40 - 4\sqrt{3} \)

- What is the area of a regular pentagon with side lengths of 5 and an apothem of 3.44?

\(43\) - Remember little Timmy? He wants to know if given a pizza with radius \(r\), what the new radius will be for a pizza with twice the area.

\( \pi r^2 = A \) so \( \pi R^2 = 2A \rightarrow \pi R^2 = 2 \pi r^2 \rightarrow R = \sqrt{2 r^2} = r \sqrt{2} \) or previous radius times the square root of 2 (approximately 41% more than previous radius). - What side lengths are needed for an equilateral triangle to have an area of 1?

Let x be the side length. The height would be \( \frac{x \sqrt{3}}{2}\) using the Pythagorean theorem. Set the area equal to 1, \( \frac{1}{2} x \frac{x \sqrt{3}}{2} = 1\) when \( x = \frac{4}{\sqrt[4]{3}}\).