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Solid geometry (surface area, volume) — ISEE Upper practice questions

Solid geometry (surface area, volume) on the ISEE Upper Mathematics Achievement test covers finding the surface area and volume of three-dimensional figures such as rectangular boxes, spheres, and cones. A student must choose the correct formula, substitute given dimensions, and solve for a missing measure such as a radius when area or volume is known. Items usually appear as short word problems about everyday objects, and the needed formula is sometimes printed in the stem. Answer choices are typically lengths, areas, or volumes, often written with π. Common traps include mixing surface area with volume, omitting a face of a closed solid, or treating a diameter as a radius.

  • Section: Mathematics Achievement 47 questions on the paper
  • 95 practice questions in the app

Sample questions

Question 1

A closed rectangular box measures 6 cm6\text{ cm} by 5 cm5\text{ cm} by 4 cm4\text{ cm}. What is the total surface area of the box?

  1. A74 cm274\text{ cm}^2
  2. B120 cm2120\text{ cm}^2
  3. C148 cm2148\text{ cm}^2
  4. D240 cm2240\text{ cm}^2
Show answer

Answer: C — 148 cm2148\text{ cm}^2

Surface area is 2(6⋅5+6⋅4+5⋅4)=2(30+24+20)=1482(6\cdot5 + 6\cdot4 + 5\cdot4) = 2(30 + 24 + 20) = 148. Omitting the factor of 2 or computing volume 6⋅5⋅4=1206\cdot5\cdot4 = 120 yields common distractors.

Question 2

A spherical ornament has surface area 100π100\pi square centimeters. What is the radius of the ornament?

(SA=4πr2)(SA=4\pi r^{2})
  1. A10 cm10\text{ cm}
  2. B5 cm5\text{ cm}
  3. C25 cm25\text{ cm}
  4. D50 cm50\text{ cm}
Show answer

Answer: B — 5 cm5\text{ cm}

4πr2=100π4\pi r^{2}=100\pi ⇒ r2=25r^{2}=25 ⇒ r=5r=5.

Question 3

An ice-cream cone has volume 48π48\pi cubic centimeters and height 9 centimeters. What is the radius of its circular base?

(V=13πr2h)(V=\dfrac{1}{3}\pi r^{2}h)
  1. A8 cm8\text{ cm}
  2. B6 cm6\text{ cm}
  3. C12 cm12\text{ cm}
  4. D4 cm4\text{ cm}
Show answer

Answer: D — 4 cm4\text{ cm}

48π=13πr2(9)48\pi=\dfrac{1}{3}\pi r^{2}(9) ⇒ 48=3r248=3r^{2} ⇒ r2=16r^{2}=16 ⇒ r=4r=4.

Practice 95 Solid geometry (surface area, volume) questions in the app

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