The diagonal length of a square is . Find the area of the square in terms of .

$\frac{t^2}{2}$

$\frac{t\sqrt{2}}2{}$

$\frac{t\sqrt{2}}{4}$

$t\sqrt{2}$

Explanation

We need to use the Pythagorean Theorem in order to solve this problem. We can write:

where is the diagonal length and is the side length. The diagonal length of the square is , so we can write:

$t^2=2s^2\Rightarrow s^2=\frac{t^2}{2}\Rightarrow Area=s^2=\frac{t^2}{2}$

A square has an area of 16 square inches. Give the diagonal of the square.

$4\sqrt{2}$

$4\sqrt{3}$

$2\sqrt{2}$

Explanation

In order to determine the length of the diagonal of a square we would use the Pythagorean Theorem. First we should find the side length:

$Side=\sqrt{area}=\sqrt{16}=4\ inches$

Now, "square" the length of one side and multiply by 2, then take the square root of that number to get the length of the diagonal:

$Diagonal=\sqrt{4^2\times 2}=4\sqrt{2}\ inches$

The volume of a cube is 1,000 cubic centimeters. Using the conversion factor 2.5 centimeters = 1 inch, give its surface area in square inches, rounding to the nearest square inch.

96 square inches

108 square inches

75 square inches

100 square inches

144 square inches

Explanation

The surface area of a cube is six times the square of its sidelength, so we find the sidelength. This is the cube root of volume 1,000, so

$\sqrt[3]{1000}$ centimeters.

To rewrite this as inches, divide by 2.5:

$\div$ inches

The surface area of the cube in square inches is

$6\times$ $4^{2}$ square inches.

The volume of a cube is 64 cubic inches. Find the side length of the cube and its surface area.

$4\ in,96\ in^{2}$

$4\ in,64\ in^{2}$

$4\ in,16\ in^{2}$

$8\ in,96\ in^{2}$

$8\ in,64\ in^{2}$

Explanation

The volume of a cube is where is the length of one edge, so is the cube root of volume:

$Volume=64\Rightarrow s=\sqrt[3]{64}=4\ in$

A cube has six faces and the surface area of a cube is . So we can write:

Surface area = $6s^2=6\times 4^2=6\times 16=96\ in^{2}$

The perimter of a square is . What is the area of the square?

Explanation

Use the perimeter to find the side length of the square.

$\text{Side}=88\div4=22$

Now, use the side length to find the area of the square.

$\text{Area}=\text{Side}^2=22^2=484$

A square is inscribed inside a circle that has an area of $12.25\pi$ square inches. What is the area of the square?

$24.5\ in^{2}$

$12.25\ in^{2}$

$28\ in^{2}$

$32.5\ in^{2}$

$49\ in^{2}$

Explanation

Start by working backwards from the area of the circle to find its diameter.

The area of a circle is $\pi r^{2}$ , and $\sqrt{12.25}=3.5$ (you can get this by trial and error pretty quickly, since you know it's between 3 and 4, and since 12.25 ends in a 5, you now that a 5 is involved), so the diameter of the circle is 7. This is also the diagonal of the square. Y

ou may remember that the legs of a right triangle (of which we have two within the square, once we draw the diagonal) are equal to the hypotenuse divided by $\sqrt{2}$ . But even if you forget this, you should recall the Pythagorean Theorem that states that $A^2+B^{2}=C^{2}$ . In this case, and are equal, so the square of each side is equal to half of $C^{2}$ , or 24.5. And the square of one side of a square is also equal to the area of the square.

A square and a circle have the same area. The circle has diameter 10. Which expression is equal to the length of one side of the square?

$5\sqrt{ \pi}$

$10 \sqrt{ \pi}$

$\pi \sqrt{5}$

$\pi \sqrt{10}$

Explanation

A circle with diameter 10 has radius half this, or 5. The area of the circle can be found using the formula

$A = \pi r ^{2}$ ,

setting :

$A = \pi \cdot 5^{2} = 25 \pi$ .

The square also has this area. The length of one side of this square is equal to the square root of this, which is

$s = \sqrt{A} = \sqrt{ 25 \pi}$

Simplify this by breaking the radicand, as follows:

$S = \sqrt{ 25 \pi} = \sqrt{ 25} \cdot \sqrt{ \pi} = 5\sqrt{ \pi}$

Right triangle 6

A square has the same perimeter as the above right triangle. Give the area of the square.

$56\frac{1}{4}$

$20 \frac{1}{4}$

Explanation

Since we know the lengths of one leg and the hypotenuse, we can calculate the length of the other leg using the Pythagorean Theorem. We can use this form:

$b = \sqrt{ c^{2} - a ^{2}}$