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Geometry - Perimeter and Area

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    Study Guide Geometry – Perimeter and Area 1. Parallelograms 1.1 Base and Height In a parallelogram: • A base is any side you choose. • The height (h) is the perpendicular distance from that base to the opposite side. Important: The height must form a right angle (90°) with the base. In the diagram: • One base is labeled b . • The non - base sides are labeled a . • The perpendicular segment is the height (h) . Figure 1

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    Study Guide 1.2 Finding the Perimeter The perimeter is the total distance around the shape. Since opposite sides of a parallelogram are equal: • Two sides are length a • Two sides are length b So the perimeter formula is: You may also see it written as: Both formulas mean the same thing. 1.3 Finding the Area To find the area of a parallelogram, think of it like a rectangle. If you rearrange parts of the parallelogram, it can form a rectangle with the same base and height. That’s why the area formula is: Where: • b is the base • h is the height The area is always the base multiplied by its corresponding height.

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    Study Guide Example 1: Solve Find the perimeter and area of the parallelogram shown. Given: • Base = 14 cm • Side = 10 cm • Height = 8 cm Figure 2 Step 1: Find the Perimeter Use the formula:

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    Study Guide Step 2: Find the Area Use the formula: Summary • Opposite sides of a parallelogram are equal. • Perimeter formula: P = 2(a + b) • Area formula: A = bh • The height must be perpendicular to the base. • You can choose any side as the base, but use the matching height. 2. Triangles Let’s look at how triangles are connected to parallelograms. Imagine triangle ABD . If we draw: • A line through B parallel to AD , and • A line through D parallel to AB , we form a parallelogram . The diagonal BD divides that parallelogram into two congruent triangles . That means: The area of triangle ABD is exactly half the area of the parallelogram.

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    Study Guide Figure 1 2.1 Finding the Area of a Triangle We know: Since a triangle is half of that parallelogram: Where: • b is the base • h is the height (perpendicular to the base) 2.2 Finding the Perimeter of a Triangle The perimeter of a triangle is simple. Just add all three sides:

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    Study Guide Example 1: Solve Find the perimeter and area of each triangle in Figures (a), (b), and (c). Figure 2 (a) Perimeter Area (b) Perimeter Area

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    Study Guide (c) Perimeter Area Example 2: Solve A triangle has: • Area = 64 cm² • Height = 16 cm Find the base. Step 1: Use the Area Formula Step 2: Multiply Both Sides by 2 Step 3: Solve for b The base of the triangle is:

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    Study Guide Summary • A triangle’s area is half the area of its associated parallelogram. • Area formula: • Perimeter formula: • The height must be perpendicular to the base. 3. Trapezoids Trapezoids are one of the most common quadrilaterals in real life. In fact, many old railroad bridges and wooden trestles were built in trapezoidal shapes because they are strong and stable. Let’s explore how to find the perimeter and area of a trapezoid. 3.1 Parts of a Trapezoid In trapezoid QRSV : • ( b 1 ) and ( b 2 ) are the bases (the parallel sides). • ( a ) and ( c ) are the legs (the nonparallel sides). • ( h ) is the height , drawn perpendicular to the bases. The perimeter is simply the sum of all four sides.

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    Study Guide Figure 1 3.2 Finding the Area of a Trapezoid Here’s a helpful idea: If you place an identical trapezoid upside down next to the original one, together they form a parallelogram . The base of that parallelogram becomes: And the height remains: So the area of the parallelogram is: But one trapezoid is only half of that parallelogram. So the area formula for a trapezoid is:

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    Study Guide Example 1: Solve Find the perimeter and area of the trapezoid shown. Given: • Left leg = 17 cm • Top base = 7 cm • Right leg = 10 cm • Bottom base = 28 cm • Height = 8 cm Figure 2 Step 1: Find the Perimeter Use the perimeter formula: Step 2: Find the Area Use the area formula:

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