Area of a Rhombus

Show your work for each problem.

1.
What is the formula for the area of a rhombus?
1. a)
2. b)
3. c)
4. d)
Answer:
The area of a rhombus is , where and are the diagonals. We divide by 2 because the diagonals create triangles, not a full rectangle.
2.
A rhombus has diagonals of 10 cm and 4 cm. What is its area in cm²?
Answer: 20
cm²
3.
Find the area of a rhombus with diagonals 6 cm and 8 cm.
Answer: 24
cm²
4.
The diagonals of a rhombus always:
1. a)Never intersect
2. b)Are equal in length
3. c)Bisect each other at right angles
4. d)Are parallel to the sides
Answer: Bisect each other at right angles
The diagonals of a rhombus bisect each other at 90° angles. This property is key to understanding why the area formula works.
5.
Calculate the area of a rhombus with diagonals cm and cm.
Answer: 30
1. What is the product of the diagonals ()? 60
2. Divide by 2 to get the area. 30
6.
A rhombus has an area of 48 cm² and one diagonal of 8 cm. Find the other diagonal in cm.
Answer: 12
From , we get cm
7.
A rhombus has an area of 70 cm². If one diagonal is 14 cm, find the length of the other diagonal.
Answer: 10
1. Write the area formula with known values: . Multiply both sides by 2. What do you get? 140
2. Now solve . What is ? 10
8.
A rhombus has diagonals of 16 cm and 12 cm. What is its area?
1. a)48 cm²
2. b)28 cm²
3. c)96 cm²
4. d)192 cm²
Answer: 96 cm²
cm². Note: 192 cm² would be wrong because it forgets to divide by 2!
9.
A kite in the shape of a rhombus has diagonals of 90 cm and 60 cm. What is the area of the kite in cm²?
Answer: 2700
cm²
10.
A rhombus-shaped tile has diagonals of 8 cm and 6 cm. How many tiles are needed to cover a surface of 1200 cm²?
Answer: 50
1. Calculate the area of one tile using . What is ? 48
2. Divide by 2 to get the area of one tile. 24
3. Divide the total area by the tile area to find the number of tiles needed. 50
11.
Two rhombuses have the same area. Rhombus A has diagonals 10 cm and 8 cm. Rhombus B has one diagonal of 20 cm. What is the other diagonal of Rhombus B?
1. a)2 cm
2. b)5 cm
3. c)4 cm
4. d)8 cm
Answer: 4 cm
Area of A = cm². For B: → cm
12.
A rhombus has diagonals of 18 cm and 24 cm. What is its area in cm²?
Answer: 216
cm²
13.
A rhombus has an area of 120 cm². If the diagonals are in the ratio 3:5, find the length of the longer diagonal.
Answer: 20
1. Let the diagonals be and . Using the area formula , simplify the right side. What is ? 15x^2
2. So . Multiply both sides by 2. What is ? 300
3. Now . Divide by 15 to find . 20
4. The longer diagonal is . Since , we have . What is simplified? 25
14.
The area of a rhombus is 120 cm². If one diagonal is twice as long as the other, find the length of the shorter diagonal in cm.
Answer: 10.95
Let and . Then . So cm.
15.
A square (which is a special rhombus) has an area of 50 cm². Find the length of each diagonal.
Answer: 10
1. In a square, both diagonals are equal. If both diagonals are , the area formula becomes . Set this equal to 50. What is ? 100
2. Take the square root of to find . 10
16.
If the diagonals of a rhombus are doubled, by what factor does the area increase?
1. a)8 times
2. b)4 times
3. c)16 times
4. d)2 times
Answer: 4 times
New area = . Original area = . Ratio = . The area increases by a factor of 4.