Contents

**Oblique Pyramid|Definition & Meaning**

**Definition**

An **oblique pyramid** is a **pyramid** with an **oblique base** and one whose **apex** is **not parallel** to the **base.** To put it another way, the **base** is **angled** or **inclined** with **respect to** the **apex.** This indicates that the **pyramid’s sides** are **not parallel** to its base and **do not intersect** at a **straight angle.**

**Overview of Pyramids **

Figure 1 – Visualization of Pyramid

A **pyramid** is a **three-dimensional geometric object** having **triangular sides meeting** at a single **point** at the **top,** known as the **apex,** and a **polygonal base. Any polygon** can **serve** as a **pyramid’s base,** but triangles and quadrilaterals are the most frequent choices. **Pyramids** are **given names** based **on** the **shape of** their **bases;** for example, a **triangular pyramid** is known as such, whereas a **quadrilateral pyramid** has a **foundation** that is **quadrilateral.**

Figure 2 – Illustration of Oblique Pyramid

A **pyramidal structure** with a **base** that **isn’t parallel** to the ground is called an **oblique pyramid.** A pyramid turned on its side can be used to describe it. An **oblique pyramid** has a **base** that can be **any quadrilateral shape,** and its **sides** are **triangles.** The **intersection** of the **sides forms** the **peak** of an oblique pyramid.

**Formulae for a Few Concepts in Oblique Pyramid**

Figure 3 – Labelling of Oblique Pyramid

**Volume**

An **oblique pyramid’s volume** is given by the formula:

**volume** = $\dfrac{\text{base area} \times \text{height}}{3}$

**Surface Area**

By **calculating** the **area** of **each** of the **triangular faces** and then **putting** them **together,** one can **determine** the **surface area** of an oblique pyramid. The **base** is the **area** of the **quadrilateral basis,** whereas the **lateral surface area** is the **area of** all the **triangular sides.**

**Lateral Area**

The **lateral area** of an oblique pyramid **is** the **sum** of the **areas** of **all** the **triangular faces.** It can be calculated using the formula

**lateral area** = $\dfrac{\text{perimeter of base} \times \text{slant height}}{2 }$

where the **slant height** is the **distance from** the **apex** to the **edge** of the base.

**Lateral Edge**

The lateral edge of an oblique pyramid is the **distance from** the **apex** to the **edge** of the base(lateral faces). It can be calculated using the Pythagorean theorem

**lateral** **edge** = $\sqrt{(\text{height})^2 + (\text{base edge})^2}$

**Slant Height**

The **slant height** of an **oblique pyramid** is the **distance from** the **apex to** the **edge** of the **base.** It can be calculated using the formula

**slant** **height** = $\sqrt{(\text{lateral edge})^2-(\text{height})^2}$

**Diagonal**

The **diagonal** of an oblique pyramid is a **line segment connecting** two **non-adjacent vertices** of the **base.** It can be calculated using the Pythagorean theorem:

**diagonal** = $\sqrt{(\text{base edge})^2 + (\text{height})^2}$

**Median**

A **median** of an oblique pyramid is a **line segment connecting** a **vertex** of the **base to** the **midpoint** of the **opposite edge.** It can be calculated using the formula

**median** = $\dfrac{\sqrt{2(\text{base edge})^2 + (\text{height})^2}} { 2}$

**The Angle Between the Lateral Edge and Base**

The **angle between** the **lateral edge** and the **base** can be calculated using the formula

**angle** = $\arctan{\left(\dfrac{\text{height}}{\text{base edge}}\right)}$

**The Angle Between the Lateral Face and Base**

The angle between a lateral face and the base can be calculated using the formula

**angle** = $\arcsin{\left(\dfrac{\text{base edge}} {\text{lateral edge}}\right)}$

**The Angle Between the Diagonal and Base**

The angle between a diagonal and the base can be calculated using the formula

**angle** = $\arccos{\left(\dfrac{\text{base edge}} {\text{diagonal}}\right)}$

**Properties of Oblique Pyramid**

Some properties of the oblique pyramid are given below:

- The
**base**of an oblique pyramid**isn’t parallel**to the ground. - An oblique pyramid’s
**base**can be**any quadrilateral shape,**but a**rectangle**is the**most typical**choice. - An oblique pyramid has
**triangle sides**that**converge**at a**single point**known as the**apex.** - The
**perpendicular distance from**an oblique pyramid’s**apex**to the**base**plane is**how tall**it is. - The
**formula (base area * height) / 3**can be**used**to get the**volume**of an oblique pyramid. - By
**calculating**the**area**of**each**of the**triangular faces**and then**putting**them**together,**one**can determine**the**surface area**of an oblique pyramid. The**base**is the**area**of the**quadrilateral basis,**whereas the**lateral surface**area is the**area**of all the**triangular sides.** - An oblique pyramid has
**vertices in**the**corners**of the**base**and an**apex.** - An oblique pyramid has
**triangles**for its**sides**and a**polygon**for its**base.** - The
**segments separating**the**vertices**on an oblique pyramid are**known**as its**edges.** - The
**faces**of an oblique pyramid are**flat surfaces**with borders enclosing them. - An oblique pyramid’s
**face count**is**influenced**by the**base’s side**count.

**Difference Between Oblique Pyramid and Right Pyramid**

Here are the main differences between an oblique pyramid and a right pyramid.

Figure 4 – Illustration of Oblique Pyramid and Right Pyramid

**Base Orientation:**Right pyramids have bases that are**perpendicular**to the**apex,**while oblique pyramids have bases that are**inclined**at an**angle**to the apex.**Shape:**A right pyramid has a**normal, symmetrical shape,**whereas an oblique pyramid has a**distorted shape**with sloping sides.**Calculating the volume of an oblique pyramid:**The**volume**of an oblique pyramid is**determined**using the**formula****V = (Bh)/3**, where**h**is the pyramid’s**slant height.**The same method is used to get the volume of a right pyramid, however here,**h**stands for the**pyramid’s height.****Calculating an oblique pyramid’s surface area:**The**surface area**of an oblique pyramid is determined using the formula**S = B + (1/2)pl**, where**l**is the**pyramid’s slant height.**A right pyramid’s surface area is computed by**adding**the**base’s area**to the**total**of its**triangle faces.****Examples from the actual world:****Oblique**pyramids are**less frequent**in real-world applications than right pyramids. Oblique pyramids are utilized less frequently than**right pyramids,**which are**employed**for a**variety**of**purposes,**such as**storage tanks**and**silos.**

**Practical Example**

**Example**

**Imagine** we have an **oblique pyramid** with a base that is a hexagon with side **lengths** of **8 cm**. The **apex** of the pyramid is **located** **12 cm** above the center of the base. **Find** the **Slant Height** of the Oblique Pyramid.

**Solution**

**First,** we need to **find** the **slant height** of the pyramid. The slant height is the distance from the apex to the base along the lateral edge of the pyramid. We can use the **Pythagorean Theorem** to find the slant height:

\[l^{2} = h^{2} + \left(\frac{B}{2}\right)^{2}\]

\[l = \sqrt{h^{2} + \left(\frac{B}{2}\right)^{2}}\]

\[l = \sqrt{12^{2} + \left(\frac{8}{2}\right)^{2}}\]

\[l = \sqrt{144 + 32}\]

\[l = \sqrt{176}\]

\[l = 13.6 \text{ cm}\]

*All mathematical drawings and images were created with GeoGebra.*