Back to Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects

Area, Volume, and Surface Area of 2D and 3D Figures

For each problem, show your work and include appropriate units in your answer.

Grade 7·21 problems·Common Core Math - Grade 7·standard·7-g-b-6

Work through problems with immediate feedback

Recall / Warm-Up

A trapezoid has parallel sides of 5 cm and 9 cm, and a height of 4 cm. What is its area?

28 cm²

56 cm²

7 cm²

36 cm²

A rectangular prism has a length of 8 m, width of 5 m, and height of 3 m. What is its volume in cubic meters?

A circle has a radius of 7 cm. Using π ≈ 3.14, what is its area?

153.86 cm²

43.96 cm²

49 cm²

21.98 cm²

Fluency Practice

$An L-shaped figure with outer dimensions 10 cm wide and 6 cm tall, with a 3 cm × 4 cm rectangle cut from the top-right corner. All key dimensions are labeled.$

An L-shaped figure is formed by starting with a 10 cm × 6 cm rectangle and removing a 3 cm × 4 cm rectangle from the top-right corner. What is the area of the L-shaped figure in square centimeters?

$A triangular prism with the triangular base showing base = 6 cm and height = 4 cm, and the prism length labeled 9 cm. Both heights are clearly distinguished.$

A triangular prism has a triangular base with a base of 6 cm and a height of 4 cm. The prism is 9 cm long. What is the volume of the prism in cubic centimeters?

A rectangular prism has dimensions 8 cm × 5 cm × 3 cm. What is its total surface area in square centimeters?

$A square pyramid with base side 6 cm. The slant height (5 cm) is labeled along one lateral face from the midpoint of the base edge to the apex.$

A square pyramid has a base of 6 cm × 6 cm and a slant height of 5 cm. What is the total surface area in square centimeters?

A trapezoidal prism has a trapezoidal base with parallel sides of 5 m and 9 m, and a base height of 4 m. The prism is 7 m long. What is the volume in cubic meters?

Word Problems

A community park has an L-shaped layout. The outer dimensions are 20 m × 15 m. A rectangular section 6 m × 8 m is removed from one corner for a playground structure.

What is the area of the park region in square meters?

A storage container is shaped like a triangular prism. Its right triangle base has legs of 3 ft and 4 ft. The container is 10 ft long.

How many cubic feet of storage space does the container hold?

$A rectangular prism (fish tank) with no lid. Labeled dimensions: 60 cm long, 30 cm wide, 35 cm tall.$

A fish tank is shaped like a rectangular prism with a length of 60 cm, a width of 30 cm, and a height of 35 cm.

How many liters of water can the tank hold when filled to the top? (1 liter = 1,000 cm³)

The tank has no lid. How many square centimeters of glass are needed to make the 5 faces of the tank?

$A 3 m × 2 m rectangular wall panel with a circular window of diameter 0.8 m cut from the center.$

A rectangular wall panel is 3 m wide and 2 m tall. A circular window with a diameter of 0.8 m is cut from the center of the panel.

What is the painted area of the panel in square meters? Use π ≈ 3.14. Round your answer to the nearest hundredth.

Error Analysis

$Two-column contrast card. Left (yellow): student's wrong work showing base height = 12 cm (prism length), giving SA = 264 cm², marked with a red X. Right (teal): correct work showing base height = 8 cm (triangle leg), giving SA = 336 cm², marked with a checkmark.$

A student computes the surface area of a triangular prism. The right triangle base has legs of 6 cm and 8 cm (hypotenuse 10 cm). The prism is 12 cm long. The student writes:

"Base height = 12 cm (prism length). Base area = (1/2)(6)(12) = 36 cm². Perimeter = 6 + 8 + 10 = 24 cm. Lateral area = 24 × 8 = 192 cm². SA = 2(36) + 192 = 264 cm²."

What error did the student make?

The student used the prism length (12 cm) as the triangle's height instead of using the triangle's own leg (8 cm) as the height, and then used the triangle's leg (8 cm) as the prism length for the lateral area.

The student used the wrong formula for the area of a right triangle.

The student forgot to include the hypotenuse in the perimeter of the triangular base.

The student multiplied the lateral area by 2 instead of computing each face separately.

$Two-column contrast card. Left (yellow): student's wrong work using vertical height 6 cm for lateral faces, giving SA = 160 cm², marked with a red X. Right (teal): correct work using slant height 10 cm, giving SA = 224 cm², marked with a checkmark.$

A student computes the surface area of a square pyramid. The base is 8 cm × 8 cm, the slant height is 10 cm, and the vertical height is 6 cm. The student writes:

"Base area = 8² = 64 cm². Each triangular face = (1/2)(8)(6) = 24 cm². SA = 64 + 4(24) = 64 + 96 = 160 cm²."

What error did the student make?

The student used the vertical height (6 cm) instead of the slant height (10 cm) for the lateral triangular faces.

The student forgot to multiply the base area by 2.

The student used the wrong number of triangular lateral faces.

The student used the wrong formula for the area of the square base.

Challenge / Extension

$A house-shaped composite solid: rectangular prism base (10 m × 8 m × 5 m) topped with a triangular prism roof (triangle base 10 m, height 3 m, depth 8 m).$

A building model is made of two parts stacked together: a rectangular prism base (length 10 m, width 8 m, height 5 m) with a triangular prism roof on top. The triangular face of the roof has a base of 10 m and a height of 3 m, and the roof extends the full 8 m width of the building.

What is the total volume of the building model in cubic meters?

Priya wants to build a closed box shaped like a triangular prism. The right triangle base has legs of 5 cm and 12 cm (hypotenuse 13 cm). The box is 20 cm long. She has 900 cm² of cardboard.

Does Priya have enough cardboard to make the box? Show your surface area calculation and explain your reasoning.