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GED Math Perimeter, Area, and the Pythagorean Theorem Summary

Checked 10/24/26

Perimeter, Area, and the Pythagorean Theorem: key points

  • Circle: area = pi x r squared; circumference = 2 x pi x r.
  • In a right triangle, a squared + b squared = c squared.
Importance
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About 4 of 21 Quantitative Problem Solving questions (20%)

Perimeter and area of rectangles, triangles, circles, polygons, and composite figures, and the Pythagorean theorem.

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Perimeter, Area, and the Pythagorean Theorem: 10 sample questions

Practice questionReviewedQuantitative Problem Solving › Perimeter, Area, and the Pythagorean Theorem★★★★☆

Question 1. A rectangular room is 12 feet wide and 15 feet long. How many square feet of carpet are needed to cover the floor?

  1. ① 180
  2. ② 54
  3. ③ 360
  4. ④ 27
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Answer: ① 180

Key point: Rectangle area A = l × w

  1. Area of a rectangle: A = length × width.
  2. A = 15 × 12 = 180 square feet.

Check: 10 × 15 = 150 and 2 × 15 = 30. 150 + 30 = 180.

Wrong choices

  • 54: 54 is the perimeter (2 × 12 + 2 × 15), the distance around the room.
  • 27: 27 adds the length and width instead of multiplying.
  • 360: 360 doubles the area; there is no need to multiply by 2.
Practice questionReviewedQuantitative Problem Solving › Perimeter, Area, and the Pythagorean Theorem★★★★☆

Question 2. The bottom of a ladder is 5 feet from a wall. The top of the ladder touches the wall 12 feet above the ground. How long is the ladder?

  1. ① 7 feet
  2. ② 13 feet
  3. ③ 169 feet
  4. ④ 17 feet
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Answer: ② 13 feet

Key point: Right triangle: a² + b² = c²

  1. The wall, the ground, and the ladder form a right triangle. The ladder is the longest side, c.
  2. a² + b² = c²: 5² + 12² = 25 + 144 = 169.
  3. c = √169 = 13 feet.

Check: 13² = 169 = 25 + 144.

Wrong choices

  • 17 feet: 17 feet adds 5 and 12; the sides must be squared first.
  • 169 feet: 169 is c², not c. Take the square root.
  • 7 feet: 7 feet subtracts 5 from 12; the ladder must be longer than either side.
Practice questionReviewedQuantitative Problem Solving › Perimeter, Area, and the Pythagorean Theorem★★★★☆

Question 3. A rectangular garden is 18 feet long and 10 feet wide. How many feet of fencing are needed to go all the way around it?

  1. ① 180 feet
  2. ② 46 feet
  3. ③ 56 feet
  4. ④ 28 feet
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Answer: ③ 56 feet

Key point: Rectangle perimeter P = 2l + 2w

  1. Perimeter: P = 2l + 2w.
  2. P = 2 × 18 + 2 × 10 = 36 + 20 = 56 feet.

Check: Walk around the garden: 18 + 10 + 18 + 10 = 56 feet.

Wrong choices

  • 28 feet: 28 feet covers only one length and one width, which is half the way around.
  • 180 feet: 180 is the area in square feet, not the distance around.
  • 46 feet: 46 feet leaves out one of the 10-foot sides.
Practice questionReviewedQuantitative Problem Solving › Perimeter, Area, and the Pythagorean Theorem★★★★☆

Question 4. A triangular sign has a base of 14 inches and a height of 9 inches. What is its area?

  1. ① 126 square inches
  2. ② 23 square inches
  3. ③ 63 square inches
  4. ④ 31.5 square inches
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Answer: ③ 63 square inches

Key point: Triangle area A = 1/2 × b × h

  1. Area of a triangle: A = 1/2 × base × height.
  2. A = 1/2 × 14 × 9 = 7 × 9 = 63 square inches.

Check: A rectangle 14 by 9 has area 126. A triangle with the same base and height is half of that: 63.

Wrong choices

  • 126 square inches: 126 square inches forgets the 1/2 in the triangle formula.
  • 23 square inches: 23 adds the base and height instead of multiplying.
  • 31.5 square inches: 31.5 square inches takes half twice (1/2 × 1/2 × 14 × 9).
Practice questionReviewedQuantitative Problem Solving › Perimeter, Area, and the Pythagorean Theorem★★★★☆

Question 5. A circular patio has a radius of 5 feet. What is its area? Use 3.14 for π.

  1. ① 78.5 square feet
  2. ② 15.7 square feet
  3. ③ 314 square feet
  4. ④ 31.4 square feet
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Answer: ① 78.5 square feet

Key point: Circle area A = π × r²

  1. Area of a circle: A = πr².
  2. r² = 5 × 5 = 25.
  3. A = 3.14 × 25 = 78.5 square feet.

Check: 3 × 25 = 75, and the answer should be a bit more than 75. 78.5 fits.

Wrong choices

  • 31.4 square feet: 31.4 is the circumference (2 × 3.14 × 5), the distance around, not the area.
  • 15.7 square feet: 15.7 multiplies π by r without squaring r.
  • 314 square feet: 314 uses the diameter (10) in place of the radius: 3.14 × 10².
Practice questionReviewedQuantitative Problem Solving › Perimeter, Area, and the Pythagorean Theorem★★★★☆

Question 6. A round table has a diameter of 4 feet. What is the distance around the edge of the table? Use 3.14 for π.

  1. ① 6.28 feet
  2. ② 50.24 feet
  3. ③ 12.56 feet
  4. ④ 25.12 feet
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Answer: ③ 12.56 feet

Key point: Circumference C = π × d = 2 × π × r

  1. Circumference: C = πd.
  2. C = 3.14 × 4 = 12.56 feet.

Check: Using C = 2πr with r = 2: 2 × 3.14 × 2 = 12.56 feet.

Wrong choices

  • 25.12 feet: 25.12 feet uses 4 as the radius; 4 feet is the diameter.
  • 50.24 feet: 50.24 is π × 4², the area if the radius were 4, not a distance around.
  • 6.28 feet: 6.28 feet uses π × r, which is only half the circumference.
Practice questionReviewedQuantitative Problem Solving › Perimeter, Area, and the Pythagorean Theorem★★★★☆

Question 7. A rectangular rug has an area of 96 square feet. Its width is 8 feet. What is the perimeter of the rug?

  1. ① 104 feet
  2. ② 20 feet
  3. ③ 12 feet
  4. ④ 40 feet
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Answer: ④ 40 feet

Key point: Find the missing side from the area, then the perimeter

  1. Length = area ÷ width = 96 ÷ 8 = 12 feet.
  2. Perimeter: 2 × 12 + 2 × 8 = 24 + 16 = 40 feet.

Check: 12 × 8 = 96 square feet, matching the given area.

Wrong choices

  • 20 feet: 20 feet is only one length plus one width (12 + 8), half the perimeter.
  • 104 feet: 104 adds the area and the width, which mixes different units.
  • 12 feet: 12 feet is the length of the rug, not the distance around it.
Practice questionReviewedQuantitative Problem Solving › Perimeter, Area, and the Pythagorean Theorem★★★★☆

Question 8. A TV screen is a rectangle with a diagonal of 50 inches and a width of 40 inches. What is the height of the screen?

  1. ① 10 inches
  2. ② 30 inches
  3. ③ 900 inches
  4. ④ 64 inches
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Answer: ② 30 inches

Key point: Missing leg: b² = c² − a²

  1. The width, height, and diagonal form a right triangle; the diagonal is the longest side c.
  2. b² = c² − a² = 50² − 40² = 2,500 − 1,600 = 900.
  3. b = √900 = 30 inches.

Check: 30² + 40² = 900 + 1,600 = 2,500 = 50².

Wrong choices

  • 10 inches: 10 inches subtracts 40 from 50 without squaring.
  • 64 inches: 64 inches adds the squares (2,500 + 1,600) and takes the root, but the diagonal is already the longest side.
  • 900 inches: 900 is the height squared; take the square root.
Practice questionReviewedQuantitative Problem Solving › Perimeter, Area, and the Pythagorean Theorem★★★★☆

Question 9. A backyard is shaped like a trapezoid. The two parallel sides are 20 feet and 30 feet long, and the distance between them is 12 feet. What is the area? (Trapezoid: A = 1/2 × h × (b₁ + b₂))

  1. ① 62 square feet
  2. ② 300 square feet
  3. ③ 240 square feet
  4. ④ 600 square feet
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Answer: ② 300 square feet

Key point: Trapezoid area = 1/2 × height × (sum of bases)

  1. Add the parallel sides: 20 + 30 = 50 feet.
  2. Multiply by the height: 50 × 12 = 600.
  3. Take half: 600 ÷ 2 = 300 square feet.

Check: The average base is (20 + 30) ÷ 2 = 25 feet, and 25 × 12 = 300.

Wrong choices

  • 600 square feet: 600 square feet forgets the 1/2 in the formula.
  • 62 square feet: 62 adds the numbers (20 + 30 + 12) instead of using the area formula.
  • 240 square feet: 240 square feet uses only the shorter side (20 × 12).
Practice questionReviewedQuantitative Problem Solving › Perimeter, Area, and the Pythagorean Theorem★★★★☆

Question 10. The circumference of a bicycle wheel is 94.2 inches. What is the diameter of the wheel? Use 3.14 for π.

  1. ① 15 inches
  2. ② 60 inches
  3. ③ 47.1 inches
  4. ④ 30 inches
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Answer: ④ 30 inches

Key point: Diameter = circumference ÷ π

  1. C = πd, so d = C ÷ π.
  2. d = 94.2 ÷ 3.14 = 30 inches.

Check: 3.14 × 30 = 94.2 inches.

Wrong choices

  • 15 inches: 15 inches is the radius (half the diameter).
  • 47.1 inches: 47.1 inches is half the circumference, not the diameter.
  • 60 inches: 60 inches doubles the diameter; 3.14 × 60 = 188.4, not 94.2.

Test yourself.

Perimeter, Area, and the Pythagorean Theorem: 5 questions →Quantitative Problem Solving: 5 questions →

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