Parallelogram Height Formula with Area and Base Explained
Understanding the Height of a Parallelogram Formula is essential for solving geometry questions in school exams, competitive tests like JEE and NEET, and for practical applications in real life. Knowing how to calculate the height using area, base, sides, or angles allows students to approach a wide range of problem types confidently.
Understanding Height in a Parallelogram
In a parallelogram, the height (or altitude) is the perpendicular distance from one side (chosen as the base) to its opposite parallel side. The height is always measured at a right angle to the base, not along the slant sides. This distinction helps avoid mistakes in geometry questions about parallelograms, rectangles, rhombuses, and related quadrilaterals.
For a clearer picture, imagine drawing a straight line from a base up to the opposite side, forming a right angle—this line is the height.
Height of a Parallelogram Formula
The standard formula for the height of a parallelogram relates the area, base, and height as follows:
Area = base × height → height = Area / base
- If you know the area and the base, divide the area by the base to get the height.
- If you know the area and a side used as the base, use that side’s length in the formula.
Where:
- Area (A): The space inside the parallelogram (square centimeters, meters, etc.)
- Base (b): The length of one side (whichever is chosen as the base)
- Height (h): Perpendicular distance from the base to the opposite side
Alternate Formula Using Sides and Angles
Sometimes, the area isn’t given directly, but the lengths of the sides and the included angle are known. Here’s how you can find the height using trigonometry:
height = side × sin(angle)
Where "side" is adjacent to the angle, and the angle is between that side and the base. This method is especially useful if you're working with sloped ("oblique") parallelograms.
Worked Examples
Example 1: Height from Area and Base
Question: A parallelogram has an area of 96 cm² and a base length of 12 cm. What is its height?
- Write the formula: height = Area / base
- Plug in the values: height = 96 / 12
- Calculate: height = 8 cm
Example 2: Height from Side and Angle
Question: A parallelogram has a side of 10 m, and the angle between this side and the base is 45°. Find the height corresponding to this base.
- Use the formula: height = side × sin(angle)
- height = 10 × sin(45°) = 10 × 0.7071 ≈ 7.071 m
Example 3: Height of a Rhombus
A rhombus has all sides equal to 15 cm and an area of 180 cm². What is its height?
- In a rhombus, use: height = Area / side
- height = 180 / 15 = 12 cm
Practice Problems
- A parallelogram has base 18 cm and area 108 cm². What is its height?
- Find the height of a parallelogram with area 350 m² and base 25 m.
- If a parallelogram has sides 20 cm each, and an angle of 30°, find the height to one of its sides.
- The sides of a parallelogram are 10 cm and 14 cm, with the included angle 60°. What is the height corresponding to the longer side?
- Given area = 72 square units, and height = 6 units. What is the length of the base?
Common Mistakes to Avoid
- Confusing "height" with slant/side—always use perpendicular height.
- Using the wrong side as the base if not specified in the question.
- Using the given angle instead of its sine value when applying the trigonometric formula.
- Forgetting units or mixing up cm, m, etc.
- Neglecting to check if the question wants the height to a particular side.
Real-World Applications
The height of a parallelogram formula is widely used in construction to find the height of slanted roofs, in architecture for area estimation, or in physics for calculating moments. It’s used wherever parallelogram-shaped surfaces, plots, or objects need precise measurement.
At Vedantu, we simplify geometry concepts like the height of a parallelogram to make problem-solving easier for students, whether in school tests or real-life applications.
Explore related geometric shapes and formulas at Area of Parallelogram and Parallelogram Properties and Types on Vedantu’s website.
Page Summary
The height of a parallelogram is the vertical distance from base to the opposite side. Use height = Area / base, or height = side × sin(angle), according to what is given. Knowing these formulas helps solve a wide variety of exam questions and real-world tasks involving parallelogram shapes. Master this concept with Vedantu’s clear explanations and practice resources for top scores in maths.
FAQs on How to Find the Height of a Parallelogram
1. What is the easiest formula for the height of a parallelogram?
The simplest formula for the height of a parallelogram is height = Area / base. This works if you already know the parallelogram's area and the length of its base.
2. How do you find the height if the area and base are given?
If you're given the area (A) and base (b) of a parallelogram, you can easily calculate its height (h) using the formula: h = A / b. Remember that the height is the perpendicular distance between the base and the opposite side.
3. How do you find the height if only sides and an angle are given?
When you know the length of one side (s) and the angle (θ) between that side and the base, you can use trigonometry to find the height (h). The formula is: h = s * sin(θ). Make sure your calculator is in degree mode if the angle is given in degrees.
4. Can you find the height without knowing the area?
Yes, you can find the height of a parallelogram without knowing the area. If you know the lengths of the two adjacent sides (a and b) and the angle (θ) between them, you can use the formula: h = a * sin(θ), where 'a' is the side forming the angle θ with the base. Alternatively, more complex methods involving vectors or diagonals may also be used.
5. What units should be used for height?
The units for height are the same as the units used for the base and area of the parallelogram. For example, if the base is measured in centimeters (cm) and the area is in square centimeters (cm²), then the height will also be in centimeters (cm).
6. What is the formula for the height of a parallelogram?
The formula for the height (h) of a parallelogram depends on what information you have. If you know the area (A) and base (b), then h = A / b. If you know a side (s) and the angle (θ) it makes with the base, then h = s * sin(θ).
7. What is a formula for a parallelogram?
Several formulas apply to parallelograms. The most common are for area (A = base × height) and perimeter (P = 2(side1 + side2)). The height formula is derived from the area formula: height = Area / base.
8. What is the height of a parallelogram whose area is 54 cm² and base is 15 cm?
To find the height, use the formula: height = Area / base. Therefore, height = 54 cm² / 15 cm = 3.6 cm.
9. How to find height of parallelogram with sides?
You can't directly find the height using only the lengths of the sides of a parallelogram. You also need either the area or an angle between two sides. If you have the area (A) and base (b), use height = A / b. If you have one side (s) and the included angle (θ), use height = s * sin(θ).
10. Height of parallelogram formula with angle?
If you know a side (s) and the angle (θ) it makes with the base, you can calculate the height (h) using trigonometry: h = s * sin(θ). This formula is useful when the area isn't provided.
