Skip to content

Quiz: Area, Volume, and Applications

Test your understanding of area and volume formulas for 2D and 3D figures, and their real-world applications.

1. What is the formula for the area of a triangle?

  1. A = bh
  2. A = (1/2)bh
  3. A = b² + h²
  4. A = (1/3)bh
Show Answer

The correct answer is B. The area of a triangle is (1/2)bh, where b is the base and h is the height (perpendicular distance from base to opposite vertex). A triangle is half of a parallelogram with the same base and height, which explains the 1/2 factor. Option A is the formula for a parallelogram. Option C resembles the Pythagorean theorem (not an area formula). Option D is used for pyramid volume, not triangle area.

Concept Tested: Area of Triangle

See: Chapter 12: Area of Triangles

2. The formula for the circumference of a circle is:

  1. C = πr²
  2. C = 2πr
  3. C = πd²
  4. C = 4πr
Show Answer

The correct answer is B. The circumference (perimeter) of a circle is C = 2πr, where r is the radius. Since the diameter d = 2r, this can also be written as C = πd. The circumference measures the distance around the circle. Option A (πr²) is the area formula for a circle, not circumference. Options C and D are incorrect formulas.

Concept Tested: Circumference of Circle

See: Chapter 12: Circles

3. What is the difference between a polyhedron and a non-polyhedron solid?

  1. Polyhedra have curved surfaces; non-polyhedra have flat faces
  2. Polyhedra have flat polygonal faces; non-polyhedra have curved surfaces
  3. Polyhedra are always larger than non-polyhedra
  4. There is no difference
Show Answer

The correct answer is B. A polyhedron is a three-dimensional solid made entirely of flat polygonal faces (like cubes, prisms, and pyramids). Non-polyhedron solids have at least one curved surface (like cylinders, cones, and spheres). Option A reverses the definitions. Option C is incorrect—size doesn't determine the classification. Option D is false—there is a clear distinction.

Concept Tested: Polyhedron vs Non-Polyhedron

See: Chapter 12: Three-Dimensional Solids

4. A rectangle has length 8 cm and width 5 cm. What is its area?

  1. 13 cm²
  2. 26 cm²
  3. 40 cm²
  4. 80 cm²
Show Answer

The correct answer is C. The area of a rectangle is A = length × width = 8 × 5 = 40 cm². Area is measured in square units (cm², m², etc.). Option A (13) is the perimeter divided by 2. Option B (26) is the perimeter. Option D (80) incorrectly doubles the area or may result from a calculation error.

Concept Tested: Area of Rectangle Application

See: Chapter 12: Area of Polygons

5. Which two formulas correctly express the volume of a cylinder with radius r and height h?

  1. V = πr²h
  2. V = (1/3)πr²h
  3. V = 2πrh
  4. V = πrh²
Show Answer

The correct answer is A. The volume of a cylinder is V = πr²h, which equals (base area) × height. The base is a circle with area πr², and when multiplied by height h, we get the volume. Option B ((1/3)πr²h) is the formula for a cone's volume, which is one-third the volume of a cylinder with the same base and height. Option C (2πrh) is the lateral surface area of a cylinder. Option D has the wrong exponents.

Concept Tested: Volume of Cylinder

See: Chapter 12: Volume Formulas

6. A trapezoid has bases of 10 cm and 14 cm, and a height of 6 cm. What is its area?

  1. 60 cm²
  2. 72 cm²
  3. 84 cm²
  4. 144 cm²
Show Answer

The correct answer is B. The area of a trapezoid is A = (1/2)(b₁ + b₂)h = (1/2)(10 + 14)(6) = (1/2)(24)(6) = 72 cm². You average the two bases and multiply by the height. Option A uses only one base. Option C adds all three measurements. Option D is the product of all three measurements.

Concept Tested: Area of Trapezoid Application

See: Chapter 12: Area of Polygons

7. Compare the volumes of a cylinder and a cone that have the same radius and height. How do they relate?

  1. The cone's volume is equal to the cylinder's volume
  2. The cone's volume is half the cylinder's volume
  3. The cone's volume is one-third the cylinder's volume
  4. The cone's volume is twice the cylinder's volume
Show Answer

The correct answer is C. A cone with the same base radius and height as a cylinder has exactly one-third the volume: V_cone = (1/3)πr²h compared to V_cylinder = πr²h. This 1:3 relationship also holds for pyramids and prisms with the same base and height. You could fit exactly three cones' worth of water into a cylinder with matching dimensions.

Concept Tested: Volume Relationship Between Cone and Cylinder

See: Chapter 12: Volume Formulas

8. A circle has a radius of 7 cm. What is its area? (Use π ≈ 3.14)

  1. 22 cm²
  2. 44 cm²
  3. 154 cm²
  4. 616 cm²
Show Answer

The correct answer is C. Using the circle area formula A = πr² = π(7)² = 49π ≈ 49 × 3.14 = 153.86 ≈ 154 cm². The area is proportional to the radius squared, so doubling the radius quadruples the area. Option A uses the radius alone. Option B uses the diameter. Option D squares the area or makes a calculation error.

Concept Tested: Area of Circle Application

See: Chapter 12: Area of Circle

9. A cube has a side length of 4 cm. What is its total surface area?

  1. 16 cm²
  2. 24 cm²
  3. 64 cm²
  4. 96 cm²
Show Answer

The correct answer is D. A cube has 6 congruent square faces. Each face has area = s² = 4² = 16 cm². Total surface area = 6 × 16 = 96 cm². The formula can also be written as SA = 6s² = 6(4)² = 96 cm². Option A is the area of one face. Option B might result from counting only some faces. Option C is the volume (4³), not surface area.

Concept Tested: Surface Area of Cube

See: Chapter 12: Surface Area

10. A sphere has a radius of 3 cm. What is its volume? (Use π ≈ 3.14 and the formula V = (4/3)πr³)

  1. 36π cm³ ≈ 113 cm³
  2. 27π cm³ ≈ 85 cm³
  3. 12π cm³ ≈ 38 cm³
  4. 9π cm³ ≈ 28 cm³
Show Answer

The correct answer is A. Using the sphere volume formula: V = (4/3)πr³ = (4/3)π(3)³ = (4/3)π(27) = (4 × 27)/3 × π = 108/3 × π = 36π ≈ 113 cm³. The sphere's volume depends on the cube of the radius, so doubling the radius multiplies the volume by 8. Option B uses only r³ without the (4/3) factor. Option C and D result from calculation errors with the exponent or fraction.

Concept Tested: Volume of Sphere Application

See: Chapter 12: Volume Formulas

Score

Check your answers above. How did you do?

  • 9-10 correct: Excellent! You have mastered area and volume calculations for 2D and 3D figures.
  • 7-8 correct: Good work! Review the formulas for circles, cylinders, cones, and spheres.
  • 5-6 correct: Study the area formulas for polygons and the volume/surface area formulas for solids.
  • Below 5: Review the chapter carefully, practice applying each formula, and pay attention to the relationships between similar shapes (like cylinder/cone).