- Basic geometry
- Course summary
- Plane and Solid Geometric Shapes: A Geometric Scavenger Hunt
- High school geometry
- Plane geometry problems

Chapter 2: Mensuration of Plane and Solid Figures. Chapter 3: Layout Procedures for Geometric Figures.

## Basic geometry

Chapter 4: Measurement and Calculation Procedures for Machinists. Chapter 5: Formulas and Calculations for Machining Operations. Chapter 7: Gear and Sprocket Calculations.

Chapter 8: Ratchets and Cam Geometry. Circles glossary Circle basics. Review articles. Right triangle trigonometry review Modeling with right triangles. Volume formulas review Solid geometry intro. Special right triangles review Special right triangles. Triangle similarity review Introduction to triangle similarity.

### Course summary

Laws of sines and cosines review Solving general triangles. Get Started Intro to Euclidean geometry. Roughly years ago, Euclid of Alexandria wrote Elements which served as the world's geometry textbook until recently. Studied by Abraham Lincoln in order to sharpen his mind and truly appreciate mathematical deduction, it is still the basis of what we consider a first year course in geometry.

### Plane and Solid Geometric Shapes: A Geometric Scavenger Hunt

Note: it will be necessary for you to know how to find the diagonal, but you don't have to memorize the formula. Continue reading for more details on this. You can find this diagonal by either using the above formula or by breaking up the figure into two flat triangles and using the pythagorean theorem for both. You can always do this is you do not want to memorize the formula or if you're afraid of mis-remembering the formula on test day. First, find the length of the diagonal hypotenuse of the base of the solid using the pythagorean theorem.

## High school geometry

Next, use that length as one of the smaller sides of a new triangle with the diagonal of the rectangular solid as the new hypotenuse. And solve for the diagonal using the pythagorean theorem again. Cubes are a special type of rectangular solid, just like squares are a special type of rectangle. The six faces on a cube's surface are also all congruent. First, identify what the question is asking you to do. Find the volume of the larger rectangle which in this case is a cube :. And divide the larger rectangular solid by the smaller to find out how many of the smaller rectangular solids can fit inside the larger:.

If you forget the formula or are afraid of messing it up come test day , you can always do it out longhand:. This is the exact same process as finding the diagonal of a rectangular solid.

Solve for the diagonal using the pythagorean theorem again. Notice that this is problem 29 an easy-medium level question , so you are given the formula.

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- Solid Geometry.
- Plane and Solid Geometry With Problems and Applications;

If this had been question 49, you would likely not have been given the formula. Pay attention, however, to exactly what the question is asking you to do. Just like with the fish tank question above, you are not being asked to fill up the whole container with water, only some of it.

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## Plane geometry problems

The radius is 12 because radius is half the diameter and the full diameter is The height is 5 because the question tells us that we are only filling up the container to 5 feet. In other words, the surface of the tube is the formula for the circumference of a circle with the additional dimension of height. Non-prism solids are shapes in three dimensions that do not have any parallel, congruent sides. A cone is similar to a cylinder, but has only one circular base instead of two. Its opposite end terminates in a point, rather than a circle.

There are two kind of cones--right cones and oblique cones. For the purposes of the ACT, you only have to concern yourself with right cones. Oblique cones will never appear on the ACT. This makes sense logically, as a cone is basically a cylinder with one base collapsed into a point.

Pyramids are geometric solids that are similar to cones, except that they have a polygon for a base and flat, triangular sides that meet at an apex. There are many types of pyramids, defined by the shape of their base and the angle of their apex, but for the sake of the SAT, you only need to concern yourself with right, square pyramids. A right, square pyramid has a square base each side has an equal length and an apex directly above the center of the base.

A sphere is essentially a 3D circle. In a circle, any straight line drawn from the center to any point on the circumference will all be equidistant. In a sphere, this radius can extend in three dimensions, so all lines from the surface of the sphere to the center of the sphere are equidistant. The most common inscribed solids on the ACT math section will be: cube inside a sphere and sphere inside a cube. You may get another shape entirely, but the basic principles of dealing with inscribed shapes will still apply.

When dealing with inscribed shapes, draw on the diagram they give you. Understand that when you are given a solid inside another solid, it is for a reason. It may look confusing to you, but the ACT will always give you enough information to solve a problem. For example, the same line will have a different meaning for each shape, and this is often the key to solving the problem. This is a trick answer designed to trap you.

This means that the diameter of the sphere will be equal to one side of the cube, because the diameter is twice the radius. But what happens when you have a sphere inside a cube? In this case, the diameter of the sphere actually becomes the diagonal of the cube.

The only straight line of the cube that touches two opposite sides of the sphere is the cube's diagonal.