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
The exterior angles (one per vertex) of a certain convex polygon with sides are obtuse. What is the value of
Tip: The sum of the measures of the exterior angles, one per vertex, of any convex polygon is
If then an equilateral triangle satisfies the condition in the prompt, as each of its exterior angles is obtuse, equalling
If the polygon has 4 sides, and each exterior angle were obtuse, then each exterior angle would have to measure more than 90 But that would mean that the sum of all exterior angles in a quadrilateral would be greater than 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).
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 and its height is decreased by What is the effect on the volume of the cylinder?
(A) It is decreased by
(B) It is decreased by
(C) It is decreased by
(D) It is increased by
(E) It is increased by
Correct Answer: B
Tip: Volume of a cylinder with base radius and height
Let the original radius be and the original height be Then the original volume of the cylinder is
is increased by and therefore, the new radius is
is decreased by and therefore, the new height is
The volume of the new cylinder is
So, the original volume has decreased by
You will get this wrong answer if you think the radius is decreased by (instead of increased) and the height is increased by (instead of decreased).
You will get this wrong answer if you subtract the two numbers in the prompt.
If you think that the change in volume is twice the change in (a decrease by 20%) plus the change in (an increase by 50%), you will get and this wrong answer.
If you don't take into account the decrease in height, you will get this wrong answer.
If you thought these examples difficult and you need to review the material, these links will help:
- Identifying Congruent Triangles
- Identifying Similar Triangles
- Comparing Similar Triangles
- Similar Triangles-Problem Solving
- Identifying Similar Polygons
- Area and Perimeter Relations in Similar Polygons
- 3D Geometry
- Composite Figures
- SAT Triangles
- SAT Right Triangles
- SAT Polygons
- SAT Circles
- SAT Congruence and Similarity
- SAT Solid Geometry
- Area of a triangle with height and base :
- Know the and the Theorems.
- The perimeter of a square with side length :
- Area of a square with side length
- Area of a rectangle with length and width
- The volume of a cube with edge length :
- The volume of a rectangular solid with length width and height
- The surface area of a cube with edge length :
- Volume of a cylinder with base radius and height
- The circumference of a circle with radius and diameter
- Area of a circle with radius
- The measure of an arc equals the measure of its central angle.
- The length of an arc with measure is
- The area of the sector formed by an arc measuring and two radii is
- SAT General Tips