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PSAT Math : Geometry

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Example Questions

Example Question #1 : How To Find The Surface Area Of A Polyhedron

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle.

The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

Possible Answers:

$6.8\textrm{ cm}$

$13.6\textrm{ cm}$

$23.8\textrm{ cm}$

$17.8\textrm{ cm}$

$11.9\textrm{ cm}$

Correct answer:

$6.8\textrm{ cm}$

Explanation:

The total surface area of the icosahedron is 400 square centimeters; since the icosahedron comprises twenty congruent faces, each has area $400 \div 20 = 20$ square centimeters.

The area of an equilateral triangle is given by the formula

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

Set A = 20 and solve for :

$\frac{s^{2}\sqrt{3}}{4} = 20$

$\frac{s^{2}\sqrt{3}}{4} \cdot \frac{4}{\sqrt{3}}= 20 \cdot \frac{4}{\sqrt{3}}$

$s^{2} = \frac{80}{ \sqrt{3}} \approx \frac{80}{1.732} \approx 46.19$

$s \approx \sqrt{46.19} \approx 6.8$ centimeters.

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Example Question #1 : How To Find The Volume Of A Polyhedron

Swimming_pool

The above depicts a rectangular swimming pool for an apartment. 80% of the pool is six feet deep, and the remaining part of the pool is four feet deep. How many cubic feet of water does the pool hold?

Possible Answers:

$10,800 \textrm{ ft}^{3}$

$9,750 \textrm{ ft}^{3}$

$9,800 \textrm{ ft}^{3}$

None of the other choices gives the correct answer.

$8,750 \textrm{ ft}^{3}$

Correct answer:

$9,800 \textrm{ ft}^{3}$

Explanation:

The cross-section of the pool is the area of its surface, which is the product of its length and its width:

$50 \times 35 = 1,750$ square feet.

Since 80% of the pool is six feet deep, this portion of the pool holds

$0.80 \times 1,750 \times 6 = 8,400$ cubic feet of water.

Since the remainder of the pool - 20% - is four feet deep, this portion of the pool holds

$0.20 \times 1,750 \times 4 = 1,400$ cubic feet of water.

Add them together: the pool holds

8,400 + 1,400 = 9,800 cubic feet of water.

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Example Question #1 : How To Find The Length Of An Edge Of A Tetrahedron

Tetra_1

Refer to the above tetrahedron, or four-faced solid. The surface area of the tetrahedron is 444. Evaluate to the nearest tenth.

Possible Answers:

$s \approx 314.0$

$s \approx 256.3$

$s \approx 21.1$

$s \approx 17.8$

$s \approx 16.0$

Correct answer:

$s \approx 16.0$

Explanation:

The tetrahedron has four faces, each of which is an equilateral triangle with sidelength . Since the total surface area is 444, each triangle has area one fourth of this, or 111. To find , set A = 111 in the formula for the area of an equilateral triangle:

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

$\frac{s^{2} \sqrt{3} }{4} =111$

$s^{2} \sqrt{3} =444$

$s^{2}=\frac{444}{ \sqrt{3} } \approx \frac{444}{ 1.732 } \approx 256.34$

$s\approx \sqrt{256.34} \approx 16.0$

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Example Question #251 : Geometry

Tetra_2

Note: Figure NOT drawn to scale.

The above triangular pyramid has volume 25. To the nearest tenth, evaluate .

Possible Answers:

Insufficient information is given to answer the problem.

$h \approx 10.8$

$h \approx 8.8$

$h \approx 13.3$

$h \approx 7.2$

Correct answer:

$h \approx 10.8$

Explanation:

We are looking for the height of the pyramid.

The base is an equilateral triangle with sidelength 4, so its area can be calculated as follows:

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

$B = \frac{4^{2} \sqrt{3} }{4}$

$B = \frac{16 \sqrt{3} }{4}$

$B = 4\sqrt{3}$

The height of a pyramid can be calculated using the fomula

$V = \frac{1}{3}Bh$

We set V = 25 and $B = 4\sqrt{3}$ and solve for :

$\frac{1}{3} \cdot 4\sqrt{3} \cdot h = 25$

$h = \frac{25 \cdot 3}{4 \sqrt{3}} = \frac{75}{4 \sqrt{3}} \approx \frac{75}{4 \cdot 1.7321} \approx 10.8$

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Example Question #252 : Geometry

Tetra_1

Note: Figure NOT drawn to scale.

Give the volume (nearest tenth) of the above triangular pyramid.

Possible Answers:

18.5

55.4

15.1

22.6

27.7

Correct answer:

18.5

Explanation:

The height of the pyramid is h = 8 . The base is an equilateral triangle with sidelength 4, so its area can be calculated as follows:

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

$B = \frac{4^{2} \sqrt{3} }{4}$

$B = \frac{16 \sqrt{3} }{4}$

$B = 4\sqrt{3}$

The volume of a pyramid can be calculated using the fomula

$V = \frac{1}{3}Bh$

$V = \frac{1}{3} \cdot 4\sqrt{3} \cdot 8 = \frac{32\sqrt{3} }{3} \approx \frac{32 \cdot 1.7321}{3} \approx 18.5$

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Example Question #252 : Geometry

A regular tetrahedron has an edge length of . What is its volume?

Possible Answers:

$\frac{32}{\sqrt{2}}$

$\frac{32}{3}$

$\frac{64}{3\sqrt{2}}$

$\frac{32}{3\sqrt{2}}$

Correct answer:

$\frac{32}{3\sqrt{2}}$

Explanation:

The volume of a tetrahedron is found with the equation $V=\frac{a^3}{6\sqrt{2}}$ , where represents the length of an edge of the tetrahedron.

Plug in 4 for the edge length and reduce as much as possible to find the answer:

$V=\frac{4^3}{6\sqrt{2}}$

$V=\frac{64}{6\sqrt{2}}$

$V=\frac{32}{3\sqrt{2}}$

The volume of the tetrahedron is $\frac{32}{3\sqrt{2}}$ .

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Example Question #1 : How To Find The Surface Area Of A Tetrahedron

A regular tetrahedron has four congruent faces, each of which is an equilateral triangle.

A given tetrahedron has edges of length six inches. Give the total surface area of the tetrahedron.

Possible Answers:

$9 \sqrt{3}\textrm{ in}^{2}$

$36 \sqrt{2}\textrm{ in}^{2}$

$9 \sqrt{2}\textrm{ in}^{2}$

$36 \textrm{ in}^{2}$

$36 \sqrt{3}\textrm{ in}^{2}$

Correct answer:

$36 \sqrt{3}\textrm{ in}^{2}$

Explanation:

The area of an equilateral triangle is given by the formula

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

Since there are four equilateral triangles that comprise the surface of the tetrahedron, the total surface area is

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

Substitute s = 6 :

$SA = s^{2}\sqrt{3} = 6^{2}\sqrt{3} = 36\sqrt{3}$ square inches.

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Example Question #1 : Tetrahedrons

Tetra_1

Give the surface area of the above tetrahedron, or four-faced solid, to the nearest tenth.

Possible Answers:

Insufficient information is given to answer the question.

114.3

98.0

84.9

69.3

Correct answer:

84.9

Explanation:

The tetrahedron has four faces, each of which is an equilateral triangle with sidelength 7. Each face has area

$A = \frac{s^{2} \sqrt{3} }{4} = \frac{7^{2} \sqrt{3} }{4} = \frac{49 \sqrt{3} }{4} \approx \frac{49 \cdot 1.7321}{4} \approx 21.22$

The total surface area is four times this, or about $4 \cdot 21.22 = 84.88$ .

Rounded, this is 84.9.

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Example Question #1 : Prisms

For a box to fit inside the cupboard, the sum of the height and the perimeter of the box must, at most, be 360 cm. If Jenn has a box that has a height of 40 cm and a length of 23 cm, what is the greatest possible width of the box?

Possible Answers:

137 cm

0.4 cm

13 cm

297 cm

207 cm

Correct answer:

137 cm

Explanation:

First we write out the equation we are given. H + (2L +2W) = 360. H = 40 and L = 23

40 + (2(23) + 2W) = 360

40 + (46 + 2W) = 360

46 + 2W = 320

2W = 274

W = 137

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Example Question #2 : Prisms

The volume of a rectangular prism is 80 cm³. The length, width, and height of the prism are each an integer number of cm. If the dimensions form three terms of an arithmetic sequence, find the average of the three dimensions.

Possible Answers:

Correct answer:

Explanation:

Method 1:

Trial and error to find a combination of factors of 80 that differ by the same amount will eventually yield 2, 5, 8. The average is 5.

Method 2:

Three terms of an arithmetic sequence can be written as x, x+d, and x+2d. Multiply these together using the distributive property to find the volume and the following equation results:

x³ + 3dx² + 2d²x - 80 = 0

Find an integer value of x that creates an integer solution for d. Try x=1 and we see the equation 1 + 3d + 2d² - 80 = 0 or 2d² + 3d -79 = 0. The determinant of this quadratic is 641, which is not a perfect square. Therefore, d is not an integer when x=1.

Try x=2 and we see the equation 8 + 12d + 4d² - 80 = 0 or d² + 3d - 18 = 0. This is easily factored to (d+6)(d-3)=0 so d=-6 or d=3. Since a negative value of d will result in negative dimensions of the prism, d must equal 3. Therefore, when substituting x=2 and d=3, the dimensions x, x+d, and x+2d become 2, 5, and 8. The average is 5.

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PSAT Math : Geometry

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : How To Find The Surface Area Of A Polyhedron

Example Question #1 : How To Find The Volume Of A Polyhedron

Example Question #1 : How To Find The Length Of An Edge Of A Tetrahedron

Example Question #251 : Geometry

Example Question #252 : Geometry

Example Question #252 : Geometry

Example Question #1 : How To Find The Surface Area Of A Tetrahedron

Example Question #1 : Tetrahedrons

Example Question #1 : Prisms

Example Question #2 : Prisms

All PSAT Math Resources

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