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

Examples

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) \(\ \ \pi s\)

Correct Answer: B

Solution:

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

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) \(\ \ 81\pi\)

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 \(BD.\)

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

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 \(DB.\)

Review

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

SAT Tips for Composite Figures

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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