Question

A design is made of a composite figure consisting of a large parallelogram and a smaller rhombus attached to one of the parallelogram's sides. The parallelogram (let's call it ABCD) has a base length (BC) of 20 cm. The rhombus (CDEF) is attached to the side CD.
Here are the specific measurements and conditions:
The length of the side CD of the parallelogram is 10 cm. Since CDEF is a rhombus, all its sides (CD, DE, EF, FC) are 10 cm long.
One of the diagonals of the rhombus (DF) measures 16 cm.
The perpendicular height of the large parallelogram (measured from the base BC up to the parallel side AD) is 15 cm.
The Challenge: Find the total area of the entire composite figure in square centimeters.

Design the figure

Answer 1

Short answer: 396 cm².

Work shown:
- Area of the parallelogram = base × perpendicular height = 20 × 15 = 300 cm².
- For the rhombus CDEF: side = 10 cm, one diagonal DF = 16 cm. In a rhombus the diagonals are perpendicular bisectors, so if one diagonal is 16 then half of it is 8 and
(half other diagonal) = sqrt(10² − 8²) = sqrt(100 − 64) = 6, so the other diagonal = 12.
Area of the rhombus = (product of diagonals)/2 = (16 × 12)/2 = 96 cm².
- Total area = 300 + 96 = 396 cm².

Important note about the description: geometrically the data as stated is inconsistent — if BC and AD are the parallel bases and their perpendicular separation (height) is 15, then any side joining a point on BC to the corresponding point on AD must be at least 15 long, so a side CD = 10 is impossible. The numeric area above treats the two pieces’ areas algebraically (base×height for the parallelogram and diagonals for the rhombus). If you want a consistent geometric drawing (or want me to adjust the height or the side length so the figure is realizable), tell me which measurement to change and I’ll give coordinates and a sketch description.