STAAR Review Volume Circles and Composite Figures

The dimensions of a rectangular prism are given in the net shown.

What is the total surface area of the rectangular prism in square inches?

Answer explanation

To determine the total surface area (total amount of space covered by the surfaces) of the rectangular prism in square inches, the student should have identified the pairs of congruent (same shape and size) rectangles in the net (two-dimensional view of a three-dimensional figure). The rectangles on the left and right have a base of 3 inches and a height of 5 inches. The area of each of these rectangles (A = bh ) is 15 square inches (3 × 5 = 1 ). The top rectangle and the third rectangle from the top have a base of 7 inches and a height of 3 inches. The area of each of these rectangles is 21 square inches (7 × 3 = 21). The second rectangle from the top and the bottom rectangle have a base of 7 inches and a height of 5 inches. The area of each of these rectangles is 35 square inches (7 × 5 = 35). The combined area of these figures, 15 + 15 + 21 + 21 + 35 + 35, is equal to the total surface area of the rectangular prism (142 square inches).

Two congruent squares and a parallelogram were used to form the figure shown.

What is the area of the figure in square meters?

Answer explanation

To determine the area (amount of space covered by a surface) of the figure, the student should have calculated the area of each shape. There are two congruent (same shape and size) squares and one parallelogram. The two squares are congruent, so the areas are equal. To find the area of each square (A = bh ) and the area of the parallelogram (A = bh ), the length of the base should be multiplied by the height (for the squares, 3 × 3 = 9, and for the parallelogram, 6 × 2.8 = 16.8). Combining the areas of the 3 shapes (9 + 9 + 16.8) results in 34.8 square meters.

The dimensions of a rectangular pyramid are shown.

What is the volume of this rectangular pyramid in cubic feet?

Answer explanation

To determine the volume (amount of three-dimensional space taken up) of the rectangular pyramid in cubic feet, the student should have used the formula for the volume of a pyramid from the Volume section of the STAAR Grade 7 Mathematics Reference Materials page within the student’s test booklet ( 1 V = Bh 3 , where V = the volume, B = the area (amount of space covered by a surface) of the base, and h = the height (vertical distance from top to bottom) of the pyramid). To determine B, the student should have multiplied 4 ft by 8 ft, resulting in 32 ft2. Then the student should have substituted the values of B = 32 and h = 6 into the volume formula 1 (V = × 32 × 6) 3 , resulting in a volume of 64 ft3.

A container in the shape of a triangular prism is used to hold a slice of pizza. The dimensions of the container are shown in the diagram.

What is the volume of the container in cubic inches?

Answer explanation

The student should have determined the volume (amount of three-dimensional space taken up) of the triangular prism (three-dimensional figure with triangular bases) in cubic inches by using the formula for the volume of a prism from the Volume section of the STAAR Grade 7 Mathematics Reference Materials page within the student’s test booklet (V = Bh , where V = volume, B is the area (amount of space covered by a surface) of the base, and h is the height (distance between the two bases, in this case triangles) of the prism). The base is a triangle, so the area of the triangle can be found with the expression 1 × 10.8 × 7.2 2 , which equals 38.88 square inches. The height of the triangular prism is 1.5 inches, so the volume of the triangular prism is determined by multiplying 38.88 square inches by 1.5 inches, which equals 58.32 cubic inches.

A rectangle, a triangle, and two congruent semicircles were used to form the figure shown.

Which measurement is closest to the area of the figure in square centimeters?

Answer explanation

To determine the area of (amount of space covered by) the figure, the student could have calculated the combined areas of the triangle, rectangle, and two semicircles (one whole circle). The base of the triangle is 6 centimeters, and the height of the triangle is 4 centimeters. The base of the rectangle is 8 centimeters, and the height of the rectangle is 6 centimeters. For each semicircle, the diameter (straight line going through the center of a circle connecting two points on the circle) is 3 centimeters. As a result, the radius (distance from the center of the circle to a point on the circle) of each semicircle is 1.5 centimeters, or half the length of the diameter. The student could have used the formulas for the areas of a triangle, rectangle, and circle from the Area section of the STAAR Grade 7 Mathematics Reference Materials page within the student’s test booklet (triangle: 1 A = b 2 h , where A = area of the triangle, b = the base, and h = the height; rectangle: A = bh , where A = area of the rectangle, b = the base, and h = the height; and circle: π 2 A = r , where A = area of the circle, π = pi (approximately 3.14), and r = radius). The student could have combined the formulas to find the area of the figure, using A = (area of the triangle) + (area of the rectangle) + (area of the whole circle), or 1 2 A = bh + (bh) + ( r ) 2 π ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ . Substituting values into the formula results in 1 2 A ≈ × 6 × 4 + (8 × 6) + (3.14 × (1.5) ) ≈ 12 + 48 + 7 = 67 2 ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ , or approximately 67 square centimeters. The rationale for the correct answer is an efficient way to solve the problem. However, other methods could be used to solve the problem correctly.

The diameter of circle X is 15 centimeters. The diameter of circle Y is 20 centimeters. Which measurement is closest to the difference between the circumference of circle X and the circumference of circle Y in centimeters?

• 15.7 cm

• 7.85 cm

• 137 cm

• 31.4 cm

Answer explanation

To determine the difference between the circumferences of (distances around the circles) circle X and circle Y, the student should have used the formula for the circumference of a circle from the Circumference section and the approximation of pi (π) from the Additional Information section of the STAAR Grade 7 Mathematics Reference Materials page within the student’s test booklet (C = πd, where C = the circumference, d = the diameter (straight line going through the center of a circle connecting two points on the circle), and π ≈ 3.14) and subtracted the values of the circumferences. To find the circumference of circle X, the student could have multiplied 3.14 by the diameter, 15 centimeters, resulting in 3.14 × 15 = 47.1 centimeters. To find the circumference of circle Y, the student could have multiplied 3.14 by the diameter, 20 centimeters, resulting in 3.14 × 20 = 62.8 centimeters. Finally, the student should have subtracted 47.1 centimeters from 62.8 centimeters, resulting in 15.7 centimeters.