SAT Composite Figures

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

A cube with edge length $s$ and a cylinder with radius $r$ and height $h$ have the same volume. If $r=s,$ which of the following is $h$ expressed in terms of $s?$

(A) $\ \ \frac{s}{\sqrt{\pi}}$

(B) $\ \ \frac{s}{\pi}$

(C) $\ \ \frac{s^2}{2\pi}$

(D) $\ \ s$

(E) $\ \ \pi s$

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Correct Answer: B

Solution:

$$\begin{array}{l c l l l} V_{\text{cylinder}} &=& V_{\text{cube}} &\quad \text{equate the volumes} &(1)\\ \pi r^2 h &=& s^3 &\quad \text{use formulas for finding volumes} &(2)\\ \pi s^2 h &=& s^3 &\quad \text{substitute}\ r=s &(3)\\ h &=& \frac{s}{\pi} &\quad \text{divide both sides by}\ s^2 \pi &(4)\\ \end{array}$$

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Incorrect Choices:

(A)
If you think that $h=s$ and you solve for $r$ in terms of $s,$ you will get this wrong answer.

(C)
Tip: Volume of a cylinder with base radius $r$ and height $h: V = \pi r^2 h.$
You will get this wrong answer if, when finding the volume of the cylinder, you use the wrong formula $V=2\pi r h.$

(D)
Tip: Volume of a cylinder with base radius $r$ and height $h: V = \pi r^2 h.$
You will get this wrong answer if, when finding the volume of the cylinder, you use the wrong formula $V=r^2 h.$

(E)
If in step $(4)$ of the solution you multiply both sides by $\pi$ instead of divide them by $\pi,$ you will get this wrong answer.

$\overline{AB}$ and $\overline{CD}$ are diameters in $\bigodot O.$ If $m\angle CAO = 50^\circ$ and $OB = 9,$what is the length of arc $BD?$

(A) $\ \ 2.5\pi$
(B) $\ \ 4\pi$
(C) $\ \ 9\pi$
(D) $\ \ 18\pi$
(E) $\ \ 81\pi$

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Correct Answer: B

Solution:

Tip: The length of an arc with measure $x^\circ$ is $\frac{x}{360}\cdot 2 \pi r.$
Tip: The measure of an arc equals the measure of its central angle.
To find the length of arc $BD,$ we must find its measure first.

We are given that $\overline{AB}$ and $\overline{CD}$ are diameters in $\bigodot O.$ Therefore, $\overline{OC}$ and $\overline{OA}$ are radii in $\bigodot O$ and $OC = OA.$ This means that $\triangle AOC$ is isosceles, with $\angle OCA = \angle OAC = 50^\circ.$

The angles in a triangle add up to $180^\circ.$ It follows that

$$\angle AOC = 180^\circ - m\angle OCA - m\angle OAC = 180^\circ-50^\circ-50^\circ = 80^\circ.$$

Notice that $\angle AOC$ and $\angle DOB$ are vertical angles. Therefore, $m\angle BOD = m\angle AOC = 80^\circ,$ and since the measure of an arc equals the measure of its central angle,

$$m \widehat BD = 80^\circ.$$

We are ready to find the length of arc $BD.$

$$\widehat{BD} = \frac{x}{360} \cdot 2 \pi r = \frac{80}{360} \cdot 2 \pi \cdot 9 = 4\pi$$

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Incorrect Choices:

(A)
If you think that $\angle BOD = 50^\circ,$ you will get this wrong answer.

(C)
This is just the radius of the circle multiplied by $\pi.$

(D)
You will get this wrong answer if you solve for the circumference of the circle, or for the area of sector $ODB,$ instead of for the length of arc $DB.$

(E)
This is the area of the circle, not the length of arc $DB.$

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

• Composite Figures
• Triangles
• Polygons
• Circles
• 3D Geometry
• SAT Triangles
• SAT Right Triangles
• SAT Polygons
• SAT Circles
• SAT Solid Geometry

• 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