Find the area of the isosceles trapezoid below by using the area formulas for rectangles and triangles.
8 cm
11 cm
4 cm
To find the area of the isosceles trapezoid, we need to divide it into a rectangle and two triangles.
First, we can find the height of the trapezoid by using the Pythagorean theorem:
h^2 = (11-8/2)^2 + 4^2
h^2 = 2.5^2 + 4^2
h^2 = 6.25 + 16
h^2 = 22.25
h = √22.25
h ≈ 4.714
Now we can calculate the area:
Area = (base1 + base2) x height / 2 + base2 x height
Area = (8 + 11) x 4.714 / 2 + 11 x 4.714
Area = 19.714 x 2.357 + 51.854
Area ≈ 98.249 cm^2
Therefore, the area of the isosceles trapezoid is approximately 98.249 cm^2.
To find the area of the isosceles trapezoid, we need to split it into two parts: a rectangle and two right triangles.
First, let's identify the sides of the trapezoid:
- The longer parallel side (base) is 11 cm.
- The shorter parallel side (top) is 4 cm.
- The two equal non-parallel sides (legs) are each 8 cm.
To find the area of the trapezoid, we need to calculate the area of the rectangle and the two triangles separately, and then add them together.
1. Area of the rectangle:
The base of the rectangle is the shorter parallel side (4 cm), and the height is the distance between the parallel sides, which is the same as the length of the legs (8 cm). The area of a rectangle is given by the formula A = length × width. Therefore, the area of the rectangle is A_rect = 4 cm × 8 cm.
2. Area of a triangle:
To calculate the area of a triangle, we can use the formula A = 1/2 × base × height. Since the triangles share the same base (8 cm), we can compute the area of one triangle and then multiply it by 2 to get the total area of both triangles.
For each triangle, the base is 8 cm, and the height is the distance from the top side to the base. The height is perpendicular to the base and can be found using the Pythagorean theorem:
height = √(legs^2 - base^2)
height = √(8 cm^2 - 4 cm^2)
Now we can compute the area of one triangle:
A_triangle = 1/2 × 8 cm × √(8 cm^2 - 4 cm^2)
Finally, to find the total area of the trapezoid, add the area of the rectangle to the combined area of the two triangles:
A_trapezoid = A_rect + 2 × A_triangle
Using the given measurements, you can substitute these values into the formulas to find the area.
To find the area of an isosceles trapezoid, we can use the formulas for rectangles and triangles.
Step 1: Find the area of the rectangle
The rectangle is formed by the base of the trapezoid, which is 8 cm, and the height of the trapezoid. Since the height is not given, we'll need to find it.
To find the height, we can use the Pythagorean Theorem. The two equal sides of the trapezoid form two right triangles. Let's call the height of the trapezoid h.
Using the Pythagorean theorem:
11^2 = h^2 + (8/2)^2
121 = h^2 + 16
h^2 = 121 - 16
h^2 = 105
h = sqrt(105)
h ≈ 10.246 cm
Now we can calculate the area of the rectangle:
Area of rectangle = base * height
Area of rectangle = 8 cm * 10.246 cm
Area of rectangle ≈ 81.968 cm^2
Step 2: Find the area of the triangle
The triangle is formed by the two equal sides of the trapezoid and the height. We already know the height (10.246 cm) and the base of the triangle, which is the same as half the base of the trapezoid (8/2 = 4 cm).
Now we can calculate the area of the triangle:
Area of triangle = (base * height) / 2
Area of triangle = (4 cm * 10.246 cm) / 2
Area of triangle ≈ 20.492 cm^2
Step 3: Find the area of the trapezoid
Finally, we can find the area of the trapezoid by adding the areas of the rectangle and the triangle:
Area of trapezoid = Area of rectangle + Area of triangle
Area of trapezoid ≈ 81.968 cm^2 + 20.492 cm^2
Area of trapezoid ≈ 102.46 cm^2
Therefore, the area of the isosceles trapezoid is approximately 102.46 cm^2.