Geometry Worksheets

Practice and master geometry concepts with helpful walkthroughs from our elite math educators, then try practice math problems to sharpen your skills.

Square Prism

By Tina Goosz

A square prism is a three-dimensional cuboid shape with squares for its top and bottom bases and four rectangles for its sides. The opposite angles and opposite sides will be congruent and parallel. There is a right square prism where the lateral faces are perpendicular to the base and an oblique prism where the lateral faces are not perpendicular to the base.

Why is this concept useful?: We use square prisms with surface area and volume with both numerical and algebraic expressions for the side lengths.
Where does this concept fit into the curriculum?: Upper Middle School / High School Math

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How can we use the concept:

Formula for volume:

a^{2} \times \ h

where a = side length of the base and h is the height of the prism.

Formula for surface area: 2 times the base surface area for both square bases plus the lateral surface area which is:

2a^{2} +4ah

To solve for volume:

a^{2} \times \ h

Substitute the side lengths into the formula. The square base will provide your side lengths and the 3 meters represents the height.

42 x 3 = 16 x 3 =

48m^{3}

To solve for surface area:

2a^{2} +4ah

Substitute the side lengths into the formula.

2( 4)^{2} \ +\ \ 4( 4)( 3) \ =\ 2( 16) \ +\ 16( 3) =\ 32\ +\ 48\ =\ 80\ m^{2}

Sample Math Problems

1.

Find the volume and surface area of the given figure (above).

Solution:

To solve for volume:

a^{2} \times \ h

3^{2} \ x\ 2\ =\ 9\ x\ 2\ =\ 18\ m^{3}

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Practice Math Problems

1.

Find the volume and surface area of the given figure (above).

2.

Find the volume and surface area of the given figure (above).

3.

Find the volume and surface area of the given figure (above).

4.

Find the volume and surface area of the given figure (above).

To solve for surface area:

2a^{2} +4ah

2( 3)^{2} +\ \ 4( 3)( 2) =\ 2( 9) \ +\ 12( 2) \ =\ 18\ +\ 24\ =\ 42\ m^{2}

2.

Find the volume and surface area of the given figure (above).

To solve for volume:

a^{2} \times \ h

2^{2} \ x\ 7=\ 4\ x\ 7\ =\ 28\ in^{3}

To solve for surface area:

2a^{2} +4ah

2( 2)^{2} \ +\ \ 4( 2)( 7) \ =\ 2( 4) \ +\ 8( 7) \ =\ 8\ +\ 56\ =\ 64\ in^{2}

3.

Find the volume and surface area of the given figure (above).

When the lengths given are algebraic expressions you will substitute in and multiply as with numeric expressions, then you will simplify into lowest terms. Your answer will remain an algebraic expression.

a^{2} \ \times \ h\

x^{2} \ \times 4x\ =\ 4x^{3}

To solve for surface area:

2a^{2} \ +\ \ 4ah

2( x)^{2} \ +\ \ 4( x)( 4x) \ =\ 2x^{2} \ +\ 4x( 4x) \ =\ 2x^{2} \ +\ 16x^{2} \ =\ 18x^{2}

4.

Find the volume and surface area of the given figure (above).

To solve for volume:

a^{2} \ \times \ h\

( 3x)^{2} \ x\ ( 3x\ +\ 4) \ =\ 9x^{2}( 3x\ +\ 4) \ =\ 27x^{2} \ +\ 36x^{2}

To solve for surface area:

2a^{2} \ +\ \ 4ah

2( 3x)^{2} \ +\ \ 4( 3x)( 3x+4) \ =\ 2\left( 9x^{2}\right) \ +\ 12x( 3x\ +\ 4) \

=\ 18x^{2} \ +\ 36x^{2} \ +\ 48x\ =\ 54x^{2} \ +\ 48x

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