Area and Volume: Practice Questions & Study Guide
Questions asking you to compute or compare areas of 2D figures and surface areas and volumes of 3D solids.
Understanding Area and Volume
Area questions on the test cover rectangles, triangles, parallelograms, trapezoids, and circles. The most frequently tested formula is the triangle area: A = (1/2)bh, where h must be the perpendicular height, not a slant side. Many errors arise when students use a slant side as the height for non-right triangles. When the height is not given directly, you often need the Pythagorean theorem or a special triangle ratio to find it first.
Composite area problems ask you to find the area of a figure that is made up of two or more standard shapes. The technique is always to decompose the figure into pieces, find each piece's area, and add (or subtract, for cut-out regions). Common compositions include a rectangle with a semicircle on top, an irregular polygon that can be split into triangles and rectangles, or a large rectangle with a triangular notch removed.
Volume questions most often involve rectangular prisms (boxes), cylinders, cones, spheres, and pyramids. The reference sheet provides all these formulas. The strategic skill is identifying the correct shape and plugging in the correct dimensions—particularly distinguishing radius from diameter for cylinders and spheres. Surface area questions (finding the total area of all faces) are less common but do appear; the approach is to enumerate every face, find each area, and sum.
Volume questions sometimes involve real-world context: a container being filled at a given rate, a comparison of two container sizes, or a scaling problem (e.g., what happens to the volume of a sphere when the radius doubles). The scaling rule is critical: if a linear dimension scales by factor k, area scales by k² and volume scales by k³.
Key Rules & Formulas
Memorize these rules — they come up directly in practice questions.
Triangle area: A = (1/2) × base × height (height must be perpendicular to the base)
Right triangle with legs 6 and 8: A = (1/2)(6)(8) = 24
Circle area: A = πr²; circumference: C = 2πr
Circle with radius 5: area = 25π ≈ 78.5; circumference = 10π ≈ 31.4
Cylinder volume: V = πr²h; cone volume: V = (1/3)πr²h
Cylinder with r = 3 and h = 10: V = π(9)(10) = 90π
Sphere volume: V = (4/3)πr³; surface area: SA = 4πr²
Sphere with r = 3: V = (4/3)π(27) = 36π
Scaling law: if all linear dimensions are multiplied by k, area scales by k² and volume scales by k³
If a box's length, width, and height are all doubled (k = 2), its volume increases by a factor of 2³ = 8
Area and Volume Practice Questions
Select an answer and click Check Answer to reveal the full explanation. Questions go from easiest to hardest.
A rectangle has a length of 12 cm and a width of 7 cm. What is the area of the rectangle in square centimeters?
Show explanation
Correct answer: C. 84
Explanation
Area of rectangle = length × width = 12 × 7 = 84 cm².
A triangle has a base of 10 inches and a height of 6 inches. What is the area of the triangle in square inches?
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Correct answer: A. 30
Explanation
Area = (1/2) × base × height = (1/2)(10)(6) = 30 square inches.
A circular pizza has a radius of 7 inches. What is the area of the pizza to the nearest square inch? (Use π ≈ 3.14)
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Correct answer: B. 154
Explanation
Area = πr² = 3.14 × 7² = 3.14 × 49 ≈ 153.86 ≈ 154 square inches.
A rectangular swimming pool is 20 meters long, 10 meters wide, and 2 meters deep. What is the volume of the pool in cubic meters?
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Correct answer: C. 400
Explanation
Volume of rectangular prism = length × width × height = 20 × 10 × 2 = 400 m³.
A cylinder has a radius of 4 cm and a height of 9 cm. What is the volume of the cylinder in terms of π?
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Correct answer: C. 144π cm³
Explanation
Volume = πr²h = π(4²)(9) = π(16)(9) = 144π cm³.
A composite figure consists of a rectangle 8 cm wide and 5 cm tall, with a semicircle of diameter 8 cm attached to the top. What is the total area of the figure? (Use π ≈ 3.14)
Show explanation
Correct answer: A. 65.12 cm²
Explanation
Rectangle area = 8 × 5 = 40 cm². The semicircle has diameter 8 cm, so radius = 4 cm. Semicircle area = (1/2)πr² = (1/2)(3.14)(16) = 25.12 cm². Total = 40 + 25.12 = 65.12 cm².
A sphere has a radius of 6 cm. What is its volume in terms of π?
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Correct answer: C. 288π cm³
Explanation
Volume of sphere = (4/3)πr³ = (4/3)π(6³) = (4/3)π(216) = 288π cm³.
A cone has a base radius of 5 cm and a slant height of 13 cm. What is the volume of the cone in terms of π?
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Correct answer: A. 100π cm³
Explanation
Find the height using the Pythagorean theorem: h = √(slant² − r²) = √(13² − 5²) = √(169 − 25) = √144 = 12 cm. Volume of cone = (1/3)πr²h = (1/3)π(25)(12) = 100π cm³.
A cylindrical tank has radius 3 feet and height 10 feet. A second tank is a cylinder with radius 6 feet and height 10 feet. How many times greater is the volume of the second tank compared to the first?
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Correct answer: C. 4
Explanation
Volume of first = π(3²)(10) = 90π. Volume of second = π(6²)(10) = 360π. Ratio = 360π / 90π = 4. The radius doubled, so the volume increased by 2² = 4 (since volume scales as r²h).
A rectangular box has dimensions l, w, and h. If all three dimensions are tripled, by what factor does the surface area increase?
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Correct answer: C. 9
Explanation
Surface area is proportional to linear dimensions squared: SA = 2(lw + lh + wh). If each dimension is tripled, each area term is multiplied by 3² = 9. Therefore, total surface area increases by a factor of 9.
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Common Mistakes to Avoid
These are the most frequent errors students make on Area and Volume questions. Knowing them in advance prevents costly point losses.
- !Using the slant height instead of the perpendicular height when computing triangle area
- !Forgetting to halve when using the triangle area formula—using A = bh instead of A = (1/2)bh
- !Confusing radius and diameter when plugging into circle or sphere formulas
- !For composite areas, forgetting to subtract the inner region when a shape has a hole or cutout
- !Misapplying the scaling law by multiplying volume by k instead of k³ when linear dimensions change
Strategy Tips: Area and Volume
For composite figures, lightly shade or label each component shape on the diagram before computing anything
Always check whether a given length is the radius or diameter—problems frequently give diameter when you need radius in the formula
Scaling questions: test with a simple specific case (e.g., unit cube doubled) to verify the k² or k³ relationship before applying it to the problem's numbers
If a 3D problem seems complex, draw the 2D cross-section—many cylinder and cone problems reduce to circle or triangle area once you identify the right slice
Other Geometry & Trigonometry Subtopics
Lines, Angles, and Triangles
Questions covering angle relationships formed by parallel lines, properties of triangles, congruence, similarity, and the triangle inequality.
Right Triangles and Trigonometry
Questions applying the Pythagorean theorem, SOH-CAH-TOA, and complementary angle trig identities to find side lengths and angle measures in right triangles.
Circles
Questions covering circle area, circumference, arc length, sector area, central and inscribed angles, and the equation of a circle in the coordinate plane.
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