GMAT Math : Rectangular Solids & Cylinders

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Example Questions

Example Question #2 : Calculating The Surface Area Of A Tetrahedron

Tetra_1

The cube in the above figure has surface area 384. Give the surface area of the tetrahedron with vertices D,E,G,H , shown in red.

Possible Answers:

128

$96 + 32 \sqrt{3}$

$128 \sqrt{3}$

$128+ 32 \sqrt{3}$

$96 + 64\sqrt{3}$

Correct answer:

$96 + 32 \sqrt{3}$

Explanation:

The surface area formula can be used to find the length of each edge of the cube:

$6s^{2}=384$

$s^{2}= 64$

s= 8

Three faces of the tetrahedron - $\bigtriangleup DHE$ , $\bigtriangleup DHG$ , $\bigtriangleup EHG$ - are right triangles with legs of length 8, so the area of each is half the product of the lengths of their legs:

$\frac{1}{2} \cdot 8 \cdot 8 =32$ .

Each triangle is isosceles, so, by the 45-45-90 Theorem, each of their hypotentuses measures $\sqrt{2}$ times a leg, or $8\sqrt{2}$ . is therefore an equilateral triangle with sidelenghth $8\sqrt{2}$ . Its area can be found as follows:

$A = \frac{s^{2}\sqrt{3}}{4}$

$= \frac{(8\sqrt{2})^{2}\sqrt{3}}{4}$

$= \frac{8^{2}\cdot (\sqrt{2})^{2} \cdot \sqrt{3}}{4}$

$= \frac{64\cdot 2 \cdot \sqrt{3}}{4}$

$= 32 \sqrt{3}$

The total surface area is

$32+32+32+ 32 \sqrt{3} = 96 + 32 \sqrt{3}$

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Example Question #1 : Calculating The Surface Area Of A Cylinder

What is the surface area of a cylinder with a radius of 7 and a height of 3?

Possible Answers:

$\dpi{100} \small 120\pi$

$\dpi{100} \small 42\pi$

$\dpi{100} \small 80\pi$

$\dpi{100} \small 140\pi$

$\dpi{100} \small 98\pi$

Correct answer:

$\dpi{100} \small 140\pi$

Explanation:

All we really need here is to remember the formula for the surface area of a cylinder.

$\dpi{100} \small SA=2\pi r^{2}+2\pi rh=2\pi \left ( 49 \right )+2\pi \left ( 7 \right )\left ( 3 \right )=98\pi +42\pi=140\pi$

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Example Question #2 : Calculating The Surface Area Of A Cylinder

The height of a cylinder is twice the circumference of its base. The radius of the base is 9 inches. What is the surface area of the cylinder?

Possible Answers:

$810 \pi \textrm{ in}^{2}$

$\left ( 162 \pi + 648 \pi^{2} \right )\textrm{ in}^{2}$

$810 \pi^{2} \textrm{ in}^{2}$

$\left ( 648 \pi +162 \pi^{2} \right )\textrm{ in}^{2}$

$1,620 \pi^{2} \textrm{ in}^{2}$

Correct answer:

$\left ( 162 \pi + 648 \pi^{2} \right )\textrm{ in}^{2}$

Explanation:

The radius of the base is 9 inches, so its circumference is $2 \pi$ times this, or $2 \pi \cdot 9 = 18 \pi$ inches. The height is twice this, or $36 \pi$ inches.

Substitute $r = 9 , h = 36 \pi$ in the formula for the surface area of the cylinder:

$A = 2 \pi r (r+h)$

$A = 2 \pi \cdot 9 \cdot (9+36\pi ) = 162 \pi + 648 \pi^{2}$ square inches

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Example Question #3 : Calculating The Surface Area Of A Cylinder

Calculate the surface area of the following cylinder.

(Not drawn to scale.)

Possible Answers:

$56\pi$

$32\pi$

$154\pi$

$88\pi$

Correct answer:

$88\pi$

Explanation:

The equation for the surface area of a cylinder is:

$SA=2\pi rh+2\pi r^2$

we plug in our values: r=4, h=7 to find the surface area

$SA=2\pi (4)(7)+2\pi (4)^2$

$SA=2 \pi (28)+32\pi$

$SA=88\pi$

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Example Question #4 : Calculating The Surface Area Of A Cylinder

Calculate the surface area of the following cylinder.

(Not drawn to scale.)

Possible Answers:

$130\pi$

$360\pi$

$288\pi$

$208\pi$

Correct answer:

$130\pi$

Explanation:

The equation for the surface area of a cylinder is

$SA=2\pi rh+2\pi r^2$

We plug in our values r=5, h=8 into the equation to find our answer.

Note: we were given the diameter of the cylinder (10), in order to find the radius we had to divide the diameter by two.

$SA=2\pi (5)(8)+2\pi (5)^2$

$SA=2\pi (40)+2\pi (25)$

$SA=130\pi$

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Example Question #5 : Calculating The Surface Area Of A Cylinder

A cylinder has a height of 9 and a radius of 4. What is the total surface area of the cylinder?

Possible Answers:

$68\pi$

$88\pi$

$132\pi$

$104\pi$

$52\pi$

Correct answer:

$104\pi$

Explanation:

We are given the height and the radius of the cylinder, which is all we need to calculate its surface area. The total surface area will be the area of the two circles on the bottom and top of the cylinder, added to the surface area of the shaft. If we imagine unfolding the shaft of the cylinder, we can see we will have a rectangle whose height is the same as that of the cylinder and whose width is the circumference of the cylinder. This means our formula for the total surface area of the cylinder will be the following:

$SA=2(\pi r^2)+2\pi rh$

$SA=2\pi (4)^2+2\pi (4)(9)$

$SA=32\pi+72\pi$

$SA=104\pi$

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Example Question #506 : Geometry

Grant is making a canister out of sheet metal. The canister will be a right cylinder with a height of 150 mm. The base of the cylinder will have a radius of $\frac{5}{\pi}$ mm. If the canister will have an open top, how many square millimeters of metal does Grant need?

Possible Answers:

$1525\text{ } mm^2$

$375+\frac{25}{\pi}\text{ } mm^2$

$1500+\frac{25}{\pi} \text{ } mm^2$

$30+\frac{25}{\pi}\text{ } mm^2$

$1500+\frac{25}{\pi}\text{ } mm^2$

Correct answer:

$1500+\frac{25}{\pi} \text{ } mm^2$

Explanation:

This question is looking for the surface area of a cylinder with only 1 base. Our surface area of a cylinder is given by:

$SA=2\pi rh+2\pi r^2$

However, because we only need 1 base, we can change it to:

$SA=2\pi rh+\pi r^2$

We know our radius and height, so simply plug them in and simplify.

$SA=2\pi \left( \frac{5}{\pi}\right)150+\pi\left(\frac{5}{\pi}\right)^2=1500+\frac{25}{\pi}$

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Example Question #6 : Calculating The Surface Area Of A Cylinder

Find the surface area of a cylinder whose height is and radius is .

Possible Answers:

$45\pi$

$18\pi$

$27\pi$

$54\pi$

Correct answer:

$54\pi$

Explanation:

To find the surface area of a cylinder, you must use the following equation.

$SA=2\pi{r}h+2\pi{r^2}$

Thus,

$SA=2\pi(3)(6)+2\pi(3^2)=54\pi$

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Example Question #2 : Cylinders

A right circular cylinder has bases of radius $\pi ^{x}$ ; its height is $\pi ^{y}$ . Give its surface area.

Possible Answers:

$\pi^{2 x+y+1}$

$\pi^{ x+2y+1}$

$2 \pi^{ x+y+1}$

$\pi ^{ 2x+1 } + \pi ^{x+y+1}$

$\pi ^{ 2y+1 } + \pi ^{x+y+1}$

Correct answer:

$\pi ^{ 2x+1 } + \pi ^{x+y+1}$

Explanation:

The surface area of a cylinder can be calculated from its radius and height as follows:

$A = \pi r ^{2} + \pi rh$

Setting $r = \pi^{x}$ and $h = \pi ^{x}$ :

$A = \pi \cdot ( \pi^{x}) ^{2} + \pi \cdot \pi^{x} \cdot \pi^{y}$

$A = \pi ^{1 } \cdot \pi^{2x} + \pi ^{1 } \cdot \pi^{x} \cdot \pi^{y}$

$A = \pi ^{1 +2x} + \pi ^{1+x+y}$ or $A = \pi ^{ 2x+1 } + \pi ^{x+y+1}$

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Example Question #1 : Calculating The Volume Of A Cylinder

What is the volume of a cone with a radius of 6 and a height of 7?

Possible Answers:

$\dpi{100} \small 42\pi$

$\dpi{100} \small 96\pi$

$\dpi{100} \small 84\pi$

$\dpi{100} \small 49\pi$

$\dpi{100} \small 36\pi$

Correct answer:

$\dpi{100} \small 84\pi$

Explanation:

The only tricky part here is remembering the formula for the volume of a cone. If you don't remember the formula for the volume of a cone, you can derive it from the volume of a cylinder. The volume of a cone is simply 1/3 the volume of the cylinder. Then,

$volume = \frac{\pi r^{2}h}{3} = \frac{\pi\cdot 6^{2}\cdot 7}{3} = 84\pi$

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GMAT Math : Rectangular Solids & Cylinders

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #2 : Calculating The Surface Area Of A Tetrahedron

Example Question #1 : Calculating The Surface Area Of A Cylinder

Example Question #2 : Calculating The Surface Area Of A Cylinder

Example Question #3 : Calculating The Surface Area Of A Cylinder

Example Question #4 : Calculating The Surface Area Of A Cylinder

Example Question #5 : Calculating The Surface Area Of A Cylinder

Example Question #506 : Geometry

Example Question #6 : Calculating The Surface Area Of A Cylinder

Example Question #2 : Cylinders

Example Question #1 : Calculating The Volume Of A Cylinder

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