# SAT Conceptual Geometry

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

- the definition of symmetry
- the shapes of three-dimensional figures -- cubes, prisms, pyramids, cylinders, cones, spheres

#### Contents

## Examples

Which three-dimensional shape can be constructed from the two-dimensional cut-out shown above?

(A) $\ \$ Rectangular prism

(B) $\ \$ Cylinder

(C) $\ \$ Cone

(D) $\ \$ Cube

(E) $\ \$ Hemisphere

Correct Answer: C

Solution:Only a cylinder, a cone, and a hemisphere have circular faces, but only a cone has a pointy tip.

Incorrect Choices:

(A),(B),(D), and(E)

The solution explains why these choices are wrong.

The sides opposite each other on a standard die always add to 7. In the diagram above, what is the sum of the numbers on all the faces that cannot be seen?

(A) $\ \ 27$

(B) $\ \ 29$

(C) $\ \ 30$

(D) $\ \ 55$

(E) $\ \ 58$

Correct Answer: D

Solution 1:One die has 6 faces. In total, all four dice have $4 \cdot 6 = 24$ faces. If each pair of opposite faces sums to 7, then since there are $\frac{24}{2}=12$ pairs of faces, the sum of all the faces will equal $12 \cdot 7 =84.$

The sum of all the faces that can be seen is (1+2+3) + (3+6) + (3+5) + (5+1) = 29.

Therefore, the sum of the faces that cannot be seen is 84-29=55.

Solution 2:Let's start with the top die. It's top shows 3. Therefore, it's bottom is 7-3=4. The side opposite the 1 is 7-1 =6, and the side opposite the 2 is 7-2=5. So, the sum of the hidden faces on the top die is 4+6+5=15.

Similarly, on the second die, the face opposite the 6 is 7-6=1, and the face opposite the 3 is 7-3=4. We know that the sum of its top and bottom is 7. So, the sum of the hidden faces on the second die is 1+4+7=12.

On the third die, the face opposite the 3 is 7-3=4, and the face opposite the 5 is 7-5=2. We know that the sum of its top and bottom is 7. So, the sum of the hidden faces on the third die is 4+2+7=13.

On the bottom die, the face opposite the 5 is 7-5=2, and the face opposite the 1 is

7-1=6. We know that the sum of its top and bottom is 7. So, the sum of the hidden faces on the last die is 2+6+7=15.The sum of all the faces that cannot be seen is 15+12+13+15=55.

Incorrect Choices:

(A)

If you forget to count the top and bottom face of each die, you will get this wrong answer.

(B)

If you calculate the sum of the faces that can be seen, you will get this wrong answer.

(C)

If you forget to count the bottom face of the top die and the top and bottom face of the other three dice, you will get this wrong answer.

(E)

If you include the top face of the top die among the faces that cannot be seen, 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 Solid Geometry

**Cite as:**SAT Conceptual Geometry.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/sat-conceptual-geometry/