Radical Radicals Trivia Quiz

Quiz Answer Key and Fun Facts

1. A square's area is determined by the formula s^2 (side squared), where s is equal to the length of one side of the square. If the square's area is 256 square inches, determine the number of inches in the length of one side of the square.

Answer: 16

Set up a formula: s^2 = 256. To get s alone, find the square root of both sides of the equation. The square root of s^2 is s, and the square root of 256 is 16. A side's length is equal to sixteen inches.

2. A rectangle's area is determined by the formula L*W, where L is equal to the rectangle's length and W is equal to the rectangle's width. If W = 2*sqrt2 inches and the area of the rectangle is 24 square inches, determine the number of inches in the length of the rectangle. Express your answer in simplest radical form. (Note: sqrt denotes a radical and will for the rest of the quiz.)

Answer: 6*sqrt2

First, set up your formula: 2*sqrt2 * L = 24. You know that L has to be x*sqrt2 because the answer has no radicals and to make a sqrt2 a whole number, you must multiply by sqrt2. sqrt2 and sqrt2 make 2 when multiplied together, so you now have 2 * x * 2 = 24 as your new equation.

It is now easy to solve: 4x=24. Divide both sides by four to get x=6. x is attached to sqrt2, so your length is 6*sqrt2 inches.

3. Determine the positive integer closest to sqrt52.

Answer: 7

There are several ways of going about this problem. One is to punch 52 into your calculator and push the square root button to see that it approximately equals 7.211, which is pretty close to seven. You can also simplify your sqrt52 to 2*sqrt13. Do this by finding the prime factorization of 52, 2*2*13. You have 2 sqrt2's that cancel to two.

Then find the square root of 13 using a calculator and multiply it by 2. You'll get about 7.211, which is close to seven. If you have perfect squares memorized, you'll know that 49 and 64 are perfect squares. 52 is closer to 49, the square of seven.

4. The perimeter of a rectangle is determined by the formula 2L + 2W, where L is the length of the rectangle and W is the width of the rectangle. Given that the length of the rectangle is 12*sqrt2 centimeters long and the perimeter of the rectangle is 30*sqrt2 centimeters, find the width of the rectangle in centimeters. Express your answer in simplest radical form.

Answer: 3*sqrt2

First, set up your formula: 2(12*sqrt2) + 2W = 30*sqrt2. Simplify your 2(12*sqrt2) to 24*sqrt2, and then subtract 24*sqrt2 from both sides to get 2W = 6*sqrt2. Divide both sides by two to find that the width is equal to 3*sqrt2.

5. The length of a diagonal of a square can be determined by the formula 2 * s^2 = c^2, where s is a side length of the square and c is the diagonal length. A square's side lengths are one. If the diagonal of this square is used as a side for another square, what is the area of the new square?

Answer: 2

First, set up your formula to find the diagonal's length: 2(1^2) = c^2. It simplifies to 2 = c^2. Find the square root of both sides to get c alone. The square root of two can simply be expressed as sqrt2, and the square root of c^2 is equal to c. You then make a square using the diagonal of length sqrt2 as a side.

The side length of the square will be sqrt2. You can then use the formula s^2 to find the area. (sqrt2)^2 = sqrt2 * sqrt2 = 2.

6. The area of a triangle can be determined by the formula bh/2, where b is the base length of the triangle and h is the height of the triangle. Given that the base length of a triangle is equal to 8*sqrt3 units and the height is equal to 4*sqrt3 units, determine the number of square units in the area of the triangle.

Answer: 48

Plug your data into the formula: (8*sqrt3)(4*sqrt3)/2. Simplify by multiplying whole numbers and radicals together. You now have (32*3)/2, which can be simplified to 96/2, and then to 48 square units.

7. You draw two squares. One has a side length of 6*sqrt5 inches, and the other has a side length that is 1/3 of the length of the first square. In square inches, what is their combined area?

Answer: 200

The length of the first square's sides is 6*sqrt5 inches. Find the area by plugging 6*sqrt5 into the formula for a square's area: (6*sqrt5)^2. Square 6 to get 36. Square sqrt5 to get 5. Multiply 36 and 5 to get 180 square inches in the area of the square.

The second square has a side length that is one-third of 6*sqrt5, which is 2*sqrt5. Square it to find the area. Square 2 to get 4 and square sqrt5 to get 5. Multiply them to get 20 square inches as the area of the second square. Add 180 and 20 together to get 200 square inches in the combined area of the squares.

8. The formula for the Pythagorean Theorem is a^2 + b^2 = c^2, where a and b are legs of a right triangle and c is the hypotenuse. Given that the length of legs a and b are 4 and 8 respectively, find the length of the hypotenuse. Express your answer in simplest radical form.

Answer: 4*sqrt5

Plug your numbers into the formula: 4^2 + 8^2 = c^2. Simplify to 16 + 64 = c^2, and then to 80 = c^2. Find the square root of both sides to get c alone. The square root of c^2 is c, and the square root of 80 is found by finding the prime factorization, which is 2 * 2 * 2 * 2 * 5. You have two pairs of twos. Each pair multiplies to two, which means you can multiply them to get 4. You have 4 and sqrt5 left, which means c is equal to 4*sqrt5 units.

9. A circle's area is determined by the formula pi * r^2, where r is the circle's radius. The radius can be determined using the formula r=d/2, where d is the diameter of the circle. Given that a circle's diameter is 6*sqrt7 meters, determine the area of the circle, in square meters. Express your answer in terms of pi.

Answer: 63*pi

The radius is equal to one-half the length of the diameter. Divide 6*sqrt7 by 2 to get 3*sqrt7, the circle's radius. Plug it in the formula to find the area of the circle: (3*sqrt7)^2. Square 3 to get 9 and sqrt7 to get 7. Then, multiply them together to get 63. You then would multiply by pi, but this problem asks for the answer in terms of pi, so your answer is 63*pi.

10. The diagonal of a cube is 12 centimeters long. What is the number of centimeters in the length of one side of the cube? Express your answer in simplest radical form. (Hint: You'll use the Pythagorean Theorem several times.)

Answer: 4*sqrt3

Let x equal the side length of a cube. The diagonal of a cube can be expressed as x^2 + (x*sqrt2)^2, where x is the length of a side and x*sqrt2 is the length of a face diagonal. To prove that x*sqrt2 is the length of a face diagonal, use the Pythagorean Theorem. x^2 + x^2 = c^2. 2*x^2 = c^2. Find the square root of both sides.

The square root of c^2 is c, and the square root of 2*x^2 is x*sqrt2. c = x*sqrt2. Now, use the Pythagorean Theorem to find the length of one side of the cube. x^2 + (x*sqrt2)^2 = 12^2. Simplify to x^2 + 2*x^2 = 144, and then to 3*x^2 = 144. Divide each side by three to get x^2 = 48, and then find the square root of both sides.

The square root of x^2 is x, and the square root of 48 can be determined by its prime factorization: 2*2*2*2*3.

There are two pairs of 2's that both multiply to 2. Together, they make 4, and you have sqrt3 left over. The length of one side of the cube is 4*sqrt3 centimeters.

Source: Author xxharryxx

This quiz was reviewed by FunTrivia editor crisw before going online.
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This quiz involves radicals, which are used in problems where irrational numbers are involved. This quiz will have you doing all types of things with radicals. NOTE: The Pythagorean Theorem is also used in this quiz.

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