# ACT Math : Solid Geometry

## Example Questions

### Example Question #5 : Finding Volume Of A Rectangular Prism

A rectangular prism has the following dimensions:

Length:

Width:

Height:

Find the volume.

Explanation:

Given that the dimensions are: , , and  and that the volume of a rectangular prism can be given by the equation:

, where  is length,  is width, and  is height, the volume can be simply solved for by substituting in the values.

This final value can be approximated to .

### Example Question #5 : How To Find The Volume Of A Prism

A rectangular box has two sides with the following lengths:

and

If it possesses a volume of , what is the area of its largest side?

49

16

21

12

28

28

Explanation:

The volume of a rectangular prism is found using the following formula:

If we substitute our known values, then we can solve for the missing side.

Divide both sides of the equation by 12.

We now know that the missing length equals 7 centimeters.

This means that the box can have sides with the following dimensions: 3cm by 4cm; 7cm by 3cm; or 7cm by 4cm. The greatest area of one side belongs to the one that is 7cm by 4cm.

### Example Question #6 : Finding Volume Of A Rectangular Prism

Solve for the volume of a prism that is 4m by 3m by 8m.

Explanation:

The volume of the rectangle

so we plug in our values and obtain

.

### Example Question #6 : How To Find The Length Of An Edge

A regular tetrahedron has a surface area of . Each face of the tetrahedron has a height of . What is the length of the base of one of the faces?

Explanation:

A regular tetrahedron has 4 triangular faces. The area of one of these faces is given by:

Because the surface area is the area of all 4 faces combined, in order to find the area for one of the faces only, we must divide the surface area by 4. We know that the surface area is , therefore:

Since we now have the area of one face, and we know the height of one face is , we can now plug these values into the original formula:

Therefore, the length of the base of one face is .

### Example Question #7 : How To Find The Length Of An Edge

What is the length of an edge of a regular tetrahedron if its surface area is 156?

Explanation:

The only given information is the surface area of the regular tetrahedron.

This is a quick problem that can be easily solved for by using the formula for the surface area of a tetrahedron:

If we substitute in the given infomation, we are left with the edge being the only unknown.

### Example Question #8 : How To Find The Length Of An Edge

What is the length of a regular tetrahedron if one face has an area of 43.3 squared units and a slant height of ?

Cannot be determined

Explanation:

The problem provides the information for the slant height and the area of one of the equilateral triangle faces.

The slant height merely refers to the height of this equilateral triangle.

Therefore, if we're given the area of a triangle and it's height, we should be able to solve for it's base. The base in this case will equate to the measurement of the edge. It's helpful to remember that in this case, because all faces are equilateral triangles, the measure of one length will equate to the length of all other edges.

We can use the equation that will allow us to solve for the area of a triangle:

where  is base length and  is height.

Substituting in the information that's been provided, we get:

### Example Question #9 : How To Find The Length Of An Edge

The volume of a regular tetrahedron is 94.8. What is the measurement of one of its edges?

Cannot be determined

Explanation:

This becomes a quick problem by just utilizing the formula for the volume of a tetrahedron.

Upon substituting the value for the volume into the formula, we are left with , which represents the edge length.

### Example Question #10 : How To Find The Length Of An Edge

A tetrahedron has a volume that is twice the surface area times the edge. What is the length of the edge? (In the answer choices,  represents edge.)

Explanation:

The problem states that the volume is:

The point of the problem is to solve for the length of the edge. Becasuse there are no numbers, the final answer will be an expression.

In order to solve for it, we will have to rearrange the formula for volume in terms of

### Example Question #1 : Tetrahedrons

Calculate the diagonal of a regular tetrahedron (all of the faces are equilateral triangles) with side length .

Explanation:

The diagonal of a shape is simply the length from a vertex to the center of the face or vertex opposite to it. With a regular tetrahedron, we have a face opposite to the vertex, and this basically amounts to calculating the height of our shape.

We know that the height of a tetrahedron is  where s is the side length, so we can put  into this formula:

which gives us the correct answer.

### Example Question #1 : How To Find The Surface Area Of A Tetrahedron

If the edge length of a tetrahedron is , what is the surface area of the tetrahedron?