The area of a rhombus is 336 square cm. If the length of one of its diagonals is 48 cm, then what is the perimeter of the rhombus ?
100 cm
Calculating Rhombus Perimeter from Area and Diagonal
This problem asks us to find the perimeter of a rhombus given its area and the length of one of its diagonals. To solve this, we need to use the properties of a rhombus, specifically the relationship between its area, diagonals, and side length.
Key Formulas and Properties of a Rhombus
A rhombus is a quadrilateral with all four sides equal in length. Important properties for this problem include:
- The area of a rhombus can be calculated using the lengths of its two diagonals, \(d_1\) and \(d_2\):
Area = \(\frac{1}{2} \times d_1 \times d_2\) - The diagonals of a rhombus bisect each other at right angles. This means they divide the rhombus into four congruent right-angled triangles.
- The side length (s) of the rhombus is the hypotenuse of these right-angled triangles. The legs of these triangles are half the lengths of the diagonals (\(\frac{d_1}{2}\) and \(\frac{d_2}{2}\)).
- The relationship between the side length and half-diagonals is given by the Pythagorean theorem: \(s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2\).
- The perimeter of a rhombus is the sum of the lengths of its four equal sides: Perimeter = \(4 \times s\).
Step-by-Step Rhombus Perimeter Calculation
We are given the area of the rhombus (336 sq cm) and the length of one diagonal (\(d_1\) = 48 cm). We need to find the perimeter.
Step 1: Find the length of the second diagonal
We use the area formula: Area = \(\frac{1}{2} \times d_1 \times d_2\).
Substitute the given values:
\[336 = \frac{1}{2} \times 48 \times d_2\]Simplify the equation:
\[336 = 24 \times d_2\]Now, solve for \(d_2\):
\[d_2 = \frac{336}{24}\]Let's perform the division:
\[d_2 = 14 \text{ cm}\]So, the length of the second diagonal is 14 cm.
Step 2: Find half the length of each diagonal
The diagonals bisect each other. Half of the first diagonal is:
\[\frac{d_1}{2} = \frac{48}{2} = 24 \text{ cm}\]Half of the second diagonal is:
\[\frac{d_2}{2} = \frac{14}{2} = 7 \text{ cm}\]These lengths (24 cm and 7 cm) are the lengths of the legs of the right-angled triangles formed inside the rhombus.
Step 3: Use the Pythagorean theorem to find the side length
The side of the rhombus (s) is the hypotenuse of a right-angled triangle with legs of length 24 cm and 7 cm. Using the Pythagorean theorem (\(a^2 + b^2 = c^2\)):
\[s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2\] \[s^2 = 24^2 + 7^2\] \[s^2 = 576 + 49\] \[s^2 = 625\]Now, find the square root to get the side length s:
\[s = \sqrt{625}\] \[s = 25 \text{ cm}\]The side length of the rhombus is 25 cm.
Step 4: Calculate the perimeter of the rhombus
The perimeter of a rhombus is \(4 \times s\). Substitute the side length we found:
\[\text{Perimeter} = 4 \times 25 \text{ cm}\] \[\text{Perimeter} = 100 \text{ cm}\]Thus, the perimeter of the rhombus is 100 cm.
Summary of Rhombus Calculation
Given: Area = 336 sq cm, \(d_1\) = 48 cm
- Calculated \(d_2\) using Area formula: \(d_2\) = 14 cm
- Calculated half diagonals: \(\frac{d_1}{2}\) = 24 cm, \(\frac{d_2}{2}\) = 7 cm
- Calculated side length (s) using Pythagorean theorem: \(s = 25\) cm
- Calculated Perimeter: \(4 \times s = 4 \times 25 = 100\) cm
Revision Table: Rhombus Formulas
| Concept | Formula | Notes |
|---|---|---|
| Area of Rhombus | \(\frac{1}{2} \times d_1 \times d_2\) | \(d_1\), \(d_2\) are diagonal lengths |
| Side length (from diagonals) | \(s = \sqrt{\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2}\) | Uses Pythagorean theorem |
| Perimeter of Rhombus | \(4 \times s\) | s is the side length |
Additional Information on Rhombus Properties
Understanding the properties of a rhombus is crucial for solving geometry problems. Here are a few more points:
- All sides are equal in length.
- Opposite angles are equal.
- Adjacent angles are supplementary (sum up to 180 degrees).
- The diagonals bisect the opposite angles.
- A square is a special type of rhombus where all angles are 90 degrees (and thus diagonals are equal).
These properties, along with the area and perimeter formulas and the Pythagorean theorem, are fundamental tools for working with rhombuses.
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