Grade 8 Composite Figures: Area and Perimeter

Introduction

Composite Figures: Area and Perimeter is an important Grade 8 math skill because students are moving from simple answers toward explaining how the math works.

In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with composite figures: area and perimeter.

What Is Composite Figures: Area and Perimeter?

Composite Figures: Area and Perimeter means measuring how much flat space a figure covers by using equal-sized square units.

The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.

Understanding Composite Figures: Area and Perimeter

Before solving, students should slow down and decide what each number, shape, unit, or label represents.

Use square units that cover the figure without gaps or overlaps.
Count rows and columns when the unit squares are arranged in an array.
Connect repeated addition to multiplication when finding area.
Break complex figures into smaller rectangles when that makes the work clearer.

Visual Models

Visual Model 1

Question: A composite figure is a rectangle (\(10\) ft by \(4\) ft) with a semicircle attached on one end (diameter \(4\) ft). What is the total area? Use \(\pi\approx3.14\).

Visual Model 1

A. \(40\) ft\(^2\)
B. \(46.28\) ft\(^2\)
C. \(52.56\) ft\(^2\)
D. \(58.84\) ft\(^2\)

Why it works: Decompose: rectangle and semicircle. Rectangle area \(=10\times4=40\) ft\(^2\). Semicircle has diameter \(4\) ft, so radius \(r=2\) ft. Semicircle area \(=\frac{1}{2}\pi r^2=\frac{1}{2}\times3.14\times2^2=6.28\) ft\(^2\). Total \(=40+6.28=46.28\) ft\(^2\).

Answer: \(46.28\) ft\(^2\)

Visual Model 2

Question: A garden has the shape of a square with side length \(12\) m with a triangular corner cut out (base \(4\) m, height \(3\) m). What is the remaining area of the garden?

Visual Model 2

A. \(138\) m\(^2\)
B. \(140\) m\(^2\)
C. \(141\) m\(^2\)
D. \(144\) m\(^2\)

Why it works: Square area \(=12\times12=144\) m\(^2\). Triangle area \(=\frac{1}{2}\times4\times3=6\) m\(^2\). Remaining area \(=144-6=138\) m\(^2\). Option D traps those who skip subtraction; option C comes from minor arithmetic error.

Answer: \(138\) m\(^2\)

Worked Examples

Example 1

Question: A trapezoid has parallel sides of \(8\) cm and \(12\) cm with a height of \(6\) cm. This trapezoid is part of a composite figure. What is its area?

Example 1

A. \(48\) cm\(^2\)
B. \(60\) cm\(^2\)
C. \(72\) cm\(^2\)
D. \(80\) cm\(^2\)

Trapezoid area \(=\frac{1}{2}(b_1+b_2)\times h=\frac{1}{2}(8+12)\times6=\frac{1}{2}\times20\times6=60\) cm\(^2\).

Answer: \(60\) cm\(^2\)

Example 2

Question: A composite figure consists of a rectangle (\(6\) m by \(5\) m) with a triangle on top (base \(6\) m, height \(2\) m). Find the total area.

Example 2

A. \(30\) m\(^2\)
B. \(36\) m\(^2\)
C. \(39\) m\(^2\)
D. \(42\) m\(^2\)

Rectangle area \(=6\times5=30\) m\(^2\).
Triangle area \(=\frac{1}{2}\times6\times2=6\) m\(^2\).
Total \(=30+6=36\) m\(^2\).

Answer: \(36\) m\(^2\)

Example 3

Question: A floor plan is an L-shaped figure made of two rectangles: one is \(10\) ft by \(6\) ft, the other is \(8\) ft by \(4\) ft. The rectangles overlap in a \(4\) ft by \(4\) ft region. What is the total floor area?

Example 3

A. \(60\) ft\(^2\)
B. \(76\) ft\(^2\)
C. \(84\) ft\(^2\)
D. \(92\) ft\(^2\)

Decompose using addition and subtraction.
First rectangle \(=10\times6=60\) ft\(^2\).
Second rectangle \(=8\times4=32\) ft\(^2\).
The overlap region is \(4\times4=16\) ft\(^2\) and must be subtracted to avoid double-counting.

Answer: \(76\) ft\(^2\)

Real-World Word Problems

Problem 1

Question: A garden is shaped like a pentagon composed of a rectangle (\(8\) ft by \(5\) ft) with a triangle on top (base \(8\) ft, height \(3\) ft). What is the perimeter if all sides are connected?

Problem 1

A. \(28\) ft
B. \(32.5\) ft
C. \(36\) ft
D. \(40\) ft

Why it works: Bottom \(8\), right side \(5\), two slanted sides \(\sqrt{4^2+3^2}=5\) each, left side \(5\). Total \(=8+5+5+5+5=28\) ft.

Answer: \(28\) ft

Problem 2

Question: A student mistakenly calculated the area of a composite figure (rectangle \(9\) m by \(4\) m plus triangle base \(9\) m, height \(3\) m) by adding \(9+4+\frac{1}{2}\times9\times3=26.5\) instead of computing the rectangle's area. What is the correct total area?

Problem 2

A. \(49.5\) m\(^2\)
B. \(49.5\) units
C. \(54\) m\(^2\)
D. \(58.5\) m\(^2\)

Why it works: Rectangle area is \(9\times4=36\) m\(^2\). Triangle area is \(\frac{1}{2}\times9\times3=13.5\) m\(^2\). The total area is \(36+13.5=49.5\) m\(^2\).

Answer: \(49.5\) m\(^2\)

Common Mistakes

Counting only the outside squares instead of all squares inside the figure.
Leaving gaps or overlaps when using unit squares.
Multiplying side lengths before checking whether the figure is a rectangle.
Forgetting to write square units with an area answer.

Strategy Tips

Trace the rectangle or figure before counting.
Use rows and columns to organize unit squares.
Write an equation after the model makes sense.
Check whether the answer needs square units.

Practice Questions

Question 1

A circular pool with radius \(4\) m has a rectangular platform (length \(8\) m, width \(3\) m) removed from its center. Find the remaining area. Use \(\pi\approx3.14\).

A. \(26.24\) m\(^2\)
B. \(40.56\) m\(^2\)
C. \(41.44\) m\(^2\)
D. \(74.12\) m\(^2\)

Question 2

A semicircle is mounted on top of a rectangle. The rectangle has dimensions \(8\) m by \(5\) m, and the semicircle has diameter \(8\) m. Find the total perimeter (outline only).

Question 2

A. \(26\) m
B. \(30.56\) m
C. \(37.56\) m
D. \(44.56\) m

Question 3

A running track is composed of a rectangle (\(100\) m by \(50\) m) with two semicircles on the short ends (diameter \(50\) m each). Find the total perimeter of the track.

Question 3

A. \(300\) m
B. \(357\) m
C. \(378.5\) m
D. \(400\) m

Question 4

A composite figure consists of a rectangle with a triangular roof. The rectangle is \(12\) ft wide and \(8\) ft tall, and the triangle has the same base and a height of \(4\) ft. What is the area?

Question 4

A. \(96\) ft\(^2\)
B. \(120\) ft\(^2\)
C. \(132\) ft\(^2\)
D. \(144\) ft\(^2\)

Question 5

An irregular polygon is decomposed into a rectangle (\(7\) in by \(4\) in) and a trapezoid (bases \(7\) in and \(3\) in, height \(2\) in). Find the total area.

Question 5

A. \(28\) in\(^2\)
B. \(38\) in\(^2\)
C. \(42\) in\(^2\)
D. \(50\) in\(^2\)

Question 6

A composite figure has a square base (\(9\) cm by \(9\) cm) with three congruent right triangles on three sides (base \(3\) cm, height \(4\) cm each). What is the total area?

Question 6

A. \(81\) cm\(^2\)
B. \(99\) cm\(^2\)
C. \(117\) cm\(^2\)
D. \(126\) cm\(^2\)

Full Answer Explanations Click to show all answers and explanations