By Carla Kirkland, Chan Cleveland

The ancient pyramids have mystified people for thousands of years, just as pyramid questions on the Praxis Core exam have mystified ill-prepared test-takers. You can avoid this dreaded curse by remembering two simple formulas for the surface area and volume of a pyramid.

The first practice question asks you to find a pyramid’s surface area, while the second question drops a pyramid on top of a cube, and asks for their composite (combined) volume.

Practice questions

  1. The following pyramid has a rectangular base. What is the surface area of the pyramid?

    praxis-core-pyramid

    A. 1,215 ft.2
    B. 1,980 ft.2
    C. 1,125 ft.2
    D. 450 ft.2
    E. 900 ft.2

    Refer to the following figure for the next question.

    praxis-core-composite

  2. The composite figure is formed by a square pyramid on top of a cube. The pyramid and cube share a base. The height of the pyramid is 9 miles, and the height of the cube is 9 miles. What is the volume of the composite figure?

    A. 729 cubic miles
    B. 972 cubic miles
    C. 81 cubic miles
    D. 648 cubic miles
    E. 1,458 cubic miles

Answers and explanations

  1. The correct answer is Choice (A).

    The surface area of a pyramid is its lateral area plus its base area, or

    PRAXIS_3501

    The base perimeter of the pyramid here is the sum of its side measures, so it’s 90 ft. The slant height is given as 17 ft. The base area is the product of the length and width of the base, so it’s 450 ft.2. With those measures, you can determine the surface area of the pyramid.”

    PRAXIS_3502

    The surface area of the pyramid is 1,215 ft.2.

  2. The correct answer is Choice (B).

    The volume of the composite figure is the sum of the volume of the cube and the volume of the pyramid. Find each separately and then add the volumes. The volume of the cube is the cube of its side measure:

    PRAXIS_3503

    The volume of the cube is 729 cubic miles.

    The volume of the pyramid is a third of its base area times its height. Its base is a square, so you can square its side measure to find the base area:

    92 = 81

    The base area of the pyramid is 81 square miles. With that and the height of the pyramid, you can find the pyramid’s volume:

    PRAXIS_3504

    Notice that the volume of the pyramid is 1/3 times the volume of the cube. That’s because they have the same base area and height. The sum of the volume of the cube and the volume of the pyramid is the sum of 729 cubic miles and 243 cubic miles:

    729 + 243 = 972

    The volume of the composite figure is 972 cubic miles.