SAT Geometry Word Problems

To successfully solve solid geometry problems on the SAT, you need to know:

the contents of the Reference Information at the beginning of each math section
the properties of triangles, polygons, and circles
how to find the perimeter and area of triangles; the perimeter and area of polygons; and the circumference and area of circles
how to identify congruent triangles
how to identify and compare similar triangles
how to identify similar polygons
how to find area and perimeter relations in similar polygons
how to find the surface area and volume of rectangular solids and cylinders

Examples

The exterior angles (one per vertex) of a certain convex polygon with \(n\) sides are obtuse. What is the value of \(n?\)

(A) \(\ \ 3\)
(B) \(\ \ 4\)
(C) \(\ \ 5\)
(D) \(\ \ 6\)
(E) \(\ \ 7\)

Solution:

Tip: The sum of the measures of the exterior angles, one per vertex, of any convex polygon is \(360^\circ.\)
If \(n=3,\) then an equilateral triangle satisfies the condition in the prompt, as each of its exterior angles is obtuse, equalling \(\frac{360^\circ}{3} = 120^\circ.\)

If the polygon has 4 sides, and each exterior angle were obtuse, then each exterior angle would have to measure more than 90\(^\circ.\) But that would mean that the sum of all exterior angles in a quadrilateral would be greater than \(4 \cdot 90 = 360^\circ.\) Therefore, the exterior angles of a 4-sided polygon cannot be obtuse. We eliminate choice (B), and by the same reasoning, we eliminate choices (C), (D), and (E).

Incorrect Choices:

If you got this problem wrong, you should review SAT Polygons.

(B), (C), (D), and (E)
The Solution explains why these choices are wrong.

The radius of a cylinder is increased by \(20 \%\) and its height is decreased by \(50 \%.\) What is the effect on the volume of the cylinder?

(A) \(\ \ \) It is decreased by \(4 \%.\)
(B) \(\ \ \) It is decreased by \(28\%.\)
(C) \(\ \ \) It is decreased by \(30\%.\)
(D) \(\ \ \) It is increased by \(10 \%.\)
(E) \(\ \ \) It is increased by \(44 \%.\)

Correct Answer: B

Solution:

Tip: Volume of a cylinder with base radius \(r\) and height \(h: V = \pi r^2 h.\)
Let the original radius be \(r\) and the original height be \(h.\) Then the original volume of the cylinder is \(V=\pi r^2h.\)

\(r\) is increased by \(10 \%\) and therefore, the new radius is \(r + \frac{20}{100} \cdot r = 1.2 r.\)

\(h\) is decreased by \(15 \%\) and therefore, the new height is \(h - \frac{50}{100} \cdot h = 0.5h.\)

The volume of the new cylinder is

\[V_{\text{new}} = \pi \cdot (1.2r)^2 \cdot 0.5h = \pi \cdot 1.44r^2 \cdot 0.5h = 0.72r^2h = 0.72V \]

So, the original volume has decreased by \(1-.72 = .28 = \frac{28}{100} = 28\%.\)

Incorrect Choices:

If you got this problem wrong, you should review SAT Solid Geometry and SAT Ratios, Proportions, and Percents.

(A)
You will get this wrong answer if you think the radius is decreased by \(20 \%\) (instead of increased) and the height is increased by \(50 \%\) (instead of decreased).

(C)
You will get this wrong answer if you subtract the two numbers in the prompt.

(D)
If you think that the change in volume is twice the change in \(r\) (a decrease by 20%) plus the change in \(h\) (an increase by 50%), you will get \(\text{change in}\ V = - 2 \cdot 20 \% + 50\% = 10\%\) and this wrong answer.

(E)
If you don't take into account the decrease in height, you will get this wrong answer.

Review

If you thought these examples difficult and you need to review the material, these links will help:

SAT Tips for Geometry Word Problems

Area of a triangle with height \(h\) and base \(b\): \(A_{\triangle} = \frac{1}{2}bh.\)

Know the \(30^\circ-60^\circ-90^\circ\) and the \(45^\circ-45^\circ-90^\circ\) Theorems.

The perimeter of a square with side length \(s\): \(P_{\square} = 4s.\)

Area of a square with side length \(s: A_{\square} = s^2.\)

Area of a rectangle with length \(l\) and width \(w: A = l\cdot w.\)

The volume of a cube with edge length \(s\): \(V = s^3.\)

The volume of a rectangular solid with length \(l,\) width \(w,\) and height \(h: V = l\cdot w \cdot h.\)

The surface area of a cube with edge length \(s\): \(SA = 6s^2.\)

Volume of a cylinder with base radius \(r\) and height \(h: V = \pi r^2 h.\)

The circumference of a circle with radius \(r\) and diameter \(d: C = 2\pi r = \pi d.\)

Area of a circle with radius \(r: A_{\bigodot} = \pi r^2.\)

The measure of an arc equals the measure of its central angle.

The length of an arc with measure \(x^\circ\) is \(\frac{x}{360}\cdot 2 \pi r.\)

The area of the sector formed by an arc measuring \(x\) and two radii is \(\frac{x}{360} \cdot \pi r^2.\)

SAT General Tips

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