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Approximate Irrationals

Grade 8 Β· Decimals Β· Worksheet 1

  1. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A circle is inscribed inside this triangle, tangent to all three sides. What is the approximate radius of this inscribed circle? Round your answer to the nearest hundredth. Answer: ______________
  2. Aroha is building a square-shaped mosaic for an art project. The area of the mosaic needs to be 63 square inches. She wants to buy a wooden frame that matches the side length of the mosaic exactly, but the store only sells frames in whole-inch lengths. What is the side length of the mosaic, in inches, rounded to the nearest whole number? Answer: ______________
  3. Olivia is planting a square-shaped garden in her backyard. She wants the garden to have an area of 91 square feet so she can grow a variety of vegetables. The store sells fencing in whole foot lengths, and she needs to buy enough to go around the entire garden. To the nearest whole foot, what is the perimeter of Olivia's garden? Answer: ______________
  4. Emma is designing a rectangular garden for her school's science project. The garden's length is √72 meters and its width is √32 meters. She needs to calculate the approximate area to determine how much soil to buy. What is the approximate area of the garden in square meters, rounded to the nearest whole number? Answer: ______________
  5. Liam is designing a circular garden with a diameter of 12 meters. He needs to calculate the exact circumference to order the right amount of decorative fencing. However, the fencing company only accepts rational approximations for their ordering system. Help Liam approximate the circumference using 3.14 for Ο€. What is the approximate circumference of his garden?
    Answer: ______________
  6. √(55) β‰ˆ ? (to the nearest tenth) Answer: ______________
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Answer Key & Explanations

Approximate Irrationals Β· Grade 8 Β· Worksheet 1

  1. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A circle is inscribed inside this triangle, tangent to all three sides. What is the approximate radius of this inscribed circle? Round your answer to the nearest hundredth. Answer: 2.00 Solution: A = (0,0) B = (6,0) C = (6,8) - AB along the x-axis from (0,0) to (6,0) β†’ length = 6 - BC vertical from (6,0) to (6,8) β†’ length = 8 - AC is the hypotenuse from (0,0) to (6,8) β†’ length = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 a = BC = 8 (opposite angle A) b = AC = 10 (opposite angle B)…
    Full step-by-step solution

    Let's go step-by-step. --- **Step 1: Understand the triangle** Vertices: A = (0,0) B = (6,0) C = (6,8) This is a right triangle with: - AB along the x-axis from (0,0) to (6,0) β†’ length = 6 - BC vertical from (6,0) to (6,8) β†’ length = 8 - AC is the hypotenuse from (0,0) to (6,8) β†’ length = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 So sides: a = BC = 8 (opposite angle A) b = AC = 10 (opposite angle B) c = AB = 6 (opposite angle C) But standard notation: Right angle at B (since AB horizontal, BC vertical). So: AB = 6 (leg) BC = 8 (leg) AC = 10 (hypotenuse) --- **Step 2: Formula for inradius of a right triangle** For a right triangle with legs a, b and hypotenuse c, the inradius r is: r = (a + b - c) / 2 Here: a = 6, b = 8, c = 10 --- **Step 3: Apply formula** r = (6 + 8 - 10) / 2 r = (14 - 10) / 2 r = 4 / 2 r = 2.00 --- **Step 4: Conclusion** The radius of the inscribed circle is 2.00. --- **Final answer:** 2.00

  2. Aroha is building a square-shaped mosaic for an art project. The area of the mosaic needs to be 63 square inches. She wants to buy a wooden frame that matches the side length of the mosaic exactly, but the store only sells frames in whole-inch lengths. What is the side length of the mosaic, in inches, rounded to the nearest whole number? Answer: 8 Solution: The area of a square is side length squared, so side length = sqrt(63). Find perfect squares near 63: 7^2 = 49 and 8^2 = 64. Since 63 is closer to 64 than to 49, sqrt(63) is approximately 7.9.
    Full step-by-step solution

    Step 1: The area of a square is side length squared, so side length = sqrt(63). Step 2: Find perfect squares near 63: 7^2 = 49 and 8^2 = 64. Step 3: Since 63 is closer to 64 than to 49, sqrt(63) is approximately 7.9. Step 4: Rounding to the nearest whole number, 7.9 rounds up to 8. Step 5: The side length is approximately 8 inches. The answer is 8.

  3. Olivia is planting a square-shaped garden in her backyard. She wants the garden to have an area of 91 square feet so she can grow a variety of vegetables. The store sells fencing in whole foot lengths, and she needs to buy enough to go around the entire garden. To the nearest whole foot, what is the perimeter of Olivia's garden? Answer: 38 Solution: The area of a square is side length squared. So side = sqrt(91). Find the two perfect squares closest to 91: 9^2 = 81 and 10^2 = 100.
    Full step-by-step solution

    Step 1: The area of a square is side length squared. So side = sqrt(91). Step 2: Find the two perfect squares closest to 91: 9^2 = 81 and 10^2 = 100. Since 91 is between 81 and 100, sqrt(91) is between 9 and 10. Step 3: Estimate sqrt(91) more precisely: 9.5^2 = 90.25, 9.6^2 = 92.16. Since 91 is closer to 90.25 than to 92.16, sqrt(91) is approximately 9.5. Step 4: The perimeter of a square is 4 times the side length: 4 * 9.5 = 38. Step 5: Since the store sells fencing in whole foot lengths, 38 feet is the amount Olivia should buy. The answer is 38.

  4. Emma is designing a rectangular garden for her school's science project. The garden's length is √72 meters and its width is √32 meters. She needs to calculate the approximate area to determine how much soil to buy. What is the approximate area of the garden in square meters, rounded to the nearest whole number? Answer: 48 Solution: Write the area formula for a rectangle: Area = length Γ— width Substitute the given values: Area = √72 Γ— √32 Use the property √a Γ— √b = √(aΓ—b): Area = √(72 Γ— 32) Multiply inside the square root: 72 Γ— 32 = 2304 Calculate the square root: √2304 = 48 Since we already have an exact value, the…
    Full step-by-step solution

    Step 1: Write the area formula for a rectangle: Area = length Γ— width Step 2: Substitute the given values: Area = √72 Γ— √32 Step 3: Use the property √a Γ— √b = √(aΓ—b): Area = √(72 Γ— 32) Step 4: Multiply inside the square root: 72 Γ— 32 = 2304 Step 5: Calculate the square root: √2304 = 48 Step 6: Since we already have an exact value, the approximate area rounded to the nearest whole number is 48 The answer is 48.

  5. Liam is designing a circular garden with a diameter of 12 meters. He needs to calculate the exact circumference to order the right amount of decorative fencing. However, the fencing company only accepts rational approximations for their ordering system. Help Liam approximate the circumference using 3.14 for Ο€. What is the approximate circumference of his garden? Answer: 37.68 meters Solution: Recall the formula for circumference. The circumference C of a circle is given by C = Ο€ Γ— d, where d is the diameter. Identify the given values.
    Full step-by-step solution

    Step 1: Recall the formula for circumference. The circumference C of a circle is given by C = Ο€ Γ— d, where d is the diameter. Step 2: Identify the given values. Diameter d = 12 meters Ο€ β‰ˆ 3.14 Step 3: Substitute the values into the formula. C = 3.14 Γ— 12 Step 4: Multiply 3.14 by 12. First, multiply 3.14 by 10: 3.14 Γ— 10 = 31.4 Then multiply 3.14 by 2: 3.14 Γ— 2 = 6.28 Now add them: 31.4 + 6.28 = 37.68 Step 5: State the final answer. The approximate circumference is 37.68 meters.

  6. √(55) β‰ˆ ? (to the nearest tenth) Answer: 7.4 Solution: Identify perfect squares near 55. 7^2 = 49 and 8^2 = 64. So √55 is between 7 and 8.
    Full step-by-step solution

    Step 1: Identify perfect squares near 55. 7^2 = 49 and 8^2 = 64. So √55 is between 7 and 8. Step 2: Since 55 is closer to 49 than to 64, the square root is closer to 7. The difference from 49 is 6, and from 64 is 9. Step 3: Try 7.4: 7.4 Γ— 7.4 = 54.76. This is 0.24 below 55. Step 4: Try 7.5: 7.5 Γ— 7.5 = 56.25. This is 1.25 above 55. Step 5: Since 7.4^2 = 54.76 is only 0.24 below 55, and 7.5^2 = 56.25 is 1.25 above, 7.4 is the better approximation to the nearest tenth. The answer is 7.4.