Math Explorer · Chapter 4
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Lesson 4.7 — Chapter 4 Exercise: Area Calculations for Plane Figures

Lesson 4.7 · Chapter exercise

Area Calculations for Plane Figures

Use the complete Chapter 4 toolbox: scale areas, compare figures, choose bases and heights, cut and recombine regions, cancel overlaps, preserve area differences, and work backward from area to length.

Grade 5 enrichment 15 questions 120 points Diagrams not necessarily to scale

Exercise directions

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Tips for this exercise

Section AQuestions 1-10 are multiple choice. Select one answer for each question. Each is worth 6 points.
Section BQuestions 11-15 are extended response. Record the final answer and show enough working to explain your method. Compare it with exercise review after trying. Each is worth 12 points.

Section A — Multiple choice

Choose one answer for each question.

10 × 6 points = 60
16 pointsNot answered

If both the base and the height of a parallelogram become 3 times their original lengths, its area becomes:

26 pointsNot answered

If both parallel bases of a trapezoid become 5 times their original lengths while its perpendicular height stays unchanged, how many times the original area is the new area?

36 pointsNot answered

In the trapezoid shown, what is the relationship between the areas of shaded Regions ① and ②?

Trapezoid with intersecting diagonals and two shaded regions The two diagonals of a trapezoid intersect. The left outer triangular region is labeled 1 and the right outer triangular region is labeled 2. ① ②
Question 3 · diagram not necessarily to scale
46 pointsNot answered

A batch of steel pipes is stacked neatly. The top layer has 5 pipes, the bottom layer has 16 pipes, and each layer has exactly 1 more pipe than the layer immediately above it. How many pipes are in the stack?

56 pointsNot answered

A trapezoid has height 4 m. Its upper and lower bases do not change. When the height is increased by 2 m, the area increases by 8 m². What was the original area of the trapezoid?

66 pointsNot answered

In the rectangle, M and N are the midpoints of the left and right sides. Which option lists all the labeled triangular regions guaranteed to have the same area as Region A?

Rectangle divided into five labeled triangles M and N are the midpoints of the left and right sides. Segments divide the rectangle into regions A, B, C, D, and E. MN A B C D E
Question 6 · M and N are side midpoints
76 pointsNot answered

The total shaded area in the two squares is 26 cm². What is the unshaded area inside the larger square?

Two adjacent squares with shaded regions A larger square stands to the left. A smaller square of side 4 centimetres is attached at the lower right. The smaller square and a triangle inside the larger square are shaded. 4 cm 4 cm
Question 7 · total shaded area is 26 cm²
86 pointsNot answered

The three shaded figures lie between the same pair of parallel lines. Which figure has the greatest area?

Parallelogram, triangle, and trapezoid between parallel lines The parallelogram has base 3, the triangle base 6, and the trapezoid bases 3 and 4. All share the same height between two parallel lines. 3 6 4 3
Question 8 · all figures share the same height
96 pointsNot answered

A shepherd has 15 fence sections, each 2 m long. Using a wall as one side and all 15 whole fence sections for the other three sides, the shepherd forms a rectangular or square sheep pen. The fence sections cannot be cut. What is the greatest possible area of the pen?

106 pointsNot answered

Each small grid square has area 1 square unit. What is the area of triangle ABC?

Triangle on a seven by seven square grid Point A is at grid coordinate 3,1; B is at 1,5; and C is at 6,6, measured from the top left. ABC
Question 10 · each small square has area 1

Section B — Extended response

Record the final answer and show your reasoning.

5 × 12 points = 60
1112 pointsNot answered

In a square, one side from a pair of opposite sides is lengthened by 17 cm and the other is shortened by 10 cm, forming the trapezoid shown. The perpendicular distance between these two parallel sides stays equal to the original square’s side length. The lower base of the trapezoid is 4 times the upper base. Find the area of the trapezoid.

A square altered into a trapezoid The original square is shown as an auxiliary figure. The top side is shortened by 10 centimetres and the bottom side is lengthened by 17 centimetres. A 10 cm 17 cm
Question 11 · original square shown as an auxiliary outline
1212 pointsNot answered

A rectangle is divided into a triangle and a trapezoid as shown. The area of the triangle is 60 cm² less than the area of the trapezoid. Find the area of the trapezoid.

A 20 by 8 rectangle divided into a triangle and a trapezoid A segment joins a point on the lower side of a rectangle to the upper-right corner. The rectangle is 20 centimetres long and 8 centimetres high. 20 cm 8 cm
Question 12 · rectangle dimensions 20 cm by 8 cm
1312 pointsNot answered

Square ABCD has side length 4 cm. The area of triangle BCF is 2 cm² greater than the area of triangle DEF. E lies beyond D on the line AD, and BE meets DC at F. Find the length DE.

Square ABCD with an external point E and a crossing line E lies to the right of D on the extension of AD. Segment BE crosses DC at F. Triangles BCF and DEF are shaded. ABCDEF
Question 13 · square side length 4 cm
1412 pointsNot answered

The area of triangle ABC equals the area of trapezoid BCDE. Find BC. Give the answer in centimetres.

A triangle above a trapezoid sharing base BC The triangle has perpendicular height 7 to BC. The trapezoid has height 4 and lower base DE equal to 9. ABCDE 9 7 4
Question 14 · heights 7 cm and 4 cm; DE = 9 cm
1512 pointsNot answered

Rectangle ABCD has length BC=12 cm and width DC=8 cm. E lies on AD, F on AB, and G on DC, with BF=CG. Segments EC and FG meet at H. The area of triangle EFC is 32 cm². Find the length HG.

Rectangle ABCD with points E, F, G, and H E lies on AD, F lies on AB, G lies on DC with BF equal to CG. Segment EC meets horizontal segment FG at H. Segments EF, FC, and EC form triangle EFC. ABCD EFGH
Question 15 · BC = 12 cm, DC = 8 cm, and BF = CG
Teaching notes
The section structure, numbering, measurements, and point allocations are preserved.Wording states the intended geometric conditions explicitly.The diagrams have been redrawn as SVGs for screen and print use.Question 9 explicitly states that all fence sections must be used whole to clarify the intended constraint.