Lesson 4: Use Properties of Rectangles, Triangles, and Trapezoids

New Concept

This lesson covers measuring geometric figures. You'll master the properties of rectangles, triangles, and trapezoids, learning to calculate their perimeter (the distance around) and area (the surface covered) using specific formulas for each shape.

What’s next

Next, you’ll apply these formulas through interactive examples and practice problems. Get ready to find the perimeter and area of rectangles, triangles, and trapezoids.

Property

Linear measure is used for measuring length (inch, foot, centimeter, meter).

Area is a measure of the region needed to cover a surface, measured in square units (square inches, square centimeters).

Volume is a measure of how much it takes to fill a container, measured in cubic units (cubic inches, cubic centimeters).

Property

• Rectangles have four sides and four right ($90^\circ$) angles.
• The lengths of opposite sides are equal.
• The perimeter, $P$, of a rectangle is the sum of twice the length and twice the width.

• The area, $A$, of a rectangle is the length times the width.

Examples

• A rectangle has a length of 15 meters and a width of 8 meters. Its perimeter is $P = 2(15) + 2(8) = 30 + 16 = 46$ meters. Its area is $A = 15 \cdot 8 = 120$ square meters.

• A rectangle has a perimeter of 60 inches and a width of 12 inches. To find the length, use $60 = 2L + 2(12)$, which simplifies to $60 = 2L + 24$. Solving for $L$ gives $36 = 2L$, so the length is 18 inches.

Property

For any triangle $\triangle ABC$, the sum of the measures of the angles is $180^\circ$.

The perimeter of a triangle is the sum of the lengths of the sides.

The area of a triangle is one-half the base, $b$, times the height, $h$.

An isosceles triangle has two sides of the same length. An equilateral triangle has three sides of equal length.

Examples

• A triangle has a base of 14 cm and a height of 10 cm. Its area is $A = \frac{1}{2}(14)(10) = 70$ square cm.

• A triangle has a perimeter of 30 feet. Two of its sides are 8 feet and 12 feet. The third side is found by solving $30 = 8 + 12 + c$, which gives $c = 10$ feet.

Property

• A trapezoid is a four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not. The parallel sides are called the bases.
• The area, $A$, of a trapezoid is $A = \frac{1}{2} h (b + B)$, where $h$ is the height, $b$ is the length of the smaller base, and $B$ is the length of the bigger base.

Examples

• A trapezoid has a height of 8 inches and bases of 10 inches and 15 inches. The area is $A = \frac{1}{2} \cdot 8 \cdot (10 + 15) = 4 \cdot 25 = 100$ square inches.

• Find the area of a trapezoid with a height of 4 feet and bases of 7.5 feet and 11.5 feet. The area is $A = \frac{1}{2} \cdot 4 \cdot (7.5 + 11.5) = 2 \cdot 19 = 38$ square feet.